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6.4: More Factoring Methods


Learning Objectives

  • Factor by Grouping
    • Identify patterns that result from multiplying two binomials and how they affect factoring by grouping
    • Factor a four term polynomial by grouping terms
  • Methods for Factoring Trinomials
    • Apply an algorithm to rewrite a trinomial as a four term polynomial
    • Use factoring by grouping to factor a trinomial
    • Use a shortcut to factor trinomials of the form (ax^2+bx+c)
    • Recognize where to place negative signs when factoring a trinomial
    • Recognize when a polynomial is a difference of squares, and how it would factor as the product of two binomials

When we learned to multiply two binomials, we found that the result, before combining like terms, was a four term polynomial, as in this example: (left(x+4 ight)left(x+2 ight)=x^{2}+2x+4x+8).

We can apply what we have learned about factoring out a common monomial to return a four term polynomial to the product of two binomials. Why would we even want to do this?

Why Should I Care?

Because it is an important step in learning techniques for factoring trinomials, such as the one you get when you simplify the product of the two binomials from above:

(egin{array}{l}left(x+4 ight)left(x+2 ight)=x^{2}+2x+4x+8=x^2+6x+8end{array})

Additionally, factoring by grouping is a technique that allows us to factor a polynomial whose terms don’t all share a GCF. In the following example, we will introduce you to the technique. Remember, one of the main reasons to factor is because it will help solve polynomial equations.

Example

Factor (a^2+3a+5a+15)
[reveal-answer q=”437455″]Show Solution[/reveal-answer]
[hidden-answer a=”437455″]

There isn’t a common factor between all four terms, so we will group the terms into pairs that will enable us to find a GCF for them. For example, we wouldn’t want to group (a^2 ext{ and }15) because they don’t share a common factor.

(left(a^2+3a ight)+left(5a+15 ight))

Find the GCF of the first pair of terms.

(egin{array}{l},,,,a^2=acdot{a},,,,3a=3cdot{a} ext{GCF}=aend{array})

Factor the GCF, a, out of the first group.

(egin{array}{r}left(acdot{a}+acdot{3} ight)+left(5a+15 ight)aleft(a+3 ight)+left(5a+15 ight)end{array})

Find the GCF of the second pair of terms.

(egin{array}{r}5a=5cdot{a}15=5cdot3 ext{GCF}=5,,,,,,,end{array})

Factor 5 out of the second group.

(egin{array}{l}aleft(a+3 ight)+left(5cdot{a}+5cdot3 ight)aleft(a+3 ight)+5left(a+3 ight)end{array})

Notice that the two terms have a common factor (left(a+3 ight)).

(aleft(a+3 ight)+5left(a+3 ight))

Factor out the common factor (left(a+3 ight)) from the two terms.

(left(a+3 ight)left(a+5 ight))

Note how the a and 5 become a binomial sum, and the other factor. This is probably the most confusing part of factoring by grouping.

Answer

(a^2+3a+5a+15=left(a+3 ight)left(a+5 ight))

[/hidden-answer]

Notice that when you factor a two term polynomial, the result is a monomial times a polynomial. But the factored form of a four-term polynomial is the product of two binomials. As we noted before, this is an important middle step in learning how to factor a three term polynomial.

This process is called the grouping technique. Broken down into individual steps, here’s how to do it (you can also follow this process in the example below).

  • Group the terms of the polynomial into pairs that share a GCF.
  • Find the greatest common factor and then use the distributive property to pull out the GCF
  • Look for the common binomial between the factored terms
  • Factor the common binomial out of the groups, the other factors will make the other binomial

Let’s try factoring a few more four-term polynomials. Note how there is a now a constant in front of the (x^2) term. We will just consider this another factor when we are finding the GCF.

Example

Factor (2x^{2}+4x+5x+10).
[reveal-answer q=”313122″]Show Solution[/reveal-answer]
[hidden-answer a=”313122″]Group terms of the polynomial into pairs.

(left(2x^{2}+4x ight)+left(5x+10 ight))

Factor out the like factor, 2x, from the first group.

(2xleft(x+2 ight)+left(5x+10 ight))

Factor out the like factor, 5, from the second group.

(2xleft(x+2 ight)+5left(x+2 ight))

Look for common factors between the factored forms of the paired terms. Here, the common factor is ((x+2)).

Factor out the common factor, (left(x+2 ight)), from both terms.

(left(2x+2 ight)left(x+5 ight))

The polynomial is now factored.

Answer

(left(2x+2 ight)left(x+5 ight))

[/hidden-answer]

Another example follows that contains subtraction. Note how we choose a positive GCF from each group of terms, and the negative signs stay.

Example

Factor (2x^{2}–3x+8x–12).
[reveal-answer q=”715080″]Show Solution[/reveal-answer]
[hidden-answer a=”715080″]Group terms into pairs.

((2x^{2}–3x)+(8x–12))

Factor the common factor, x, out of the first group and the common factor, 4, out of the second group.

(xleft(2x–3 ight)+4left(2x–3 ight))

Factor out the common factor, (left(2x–3 ight)), from both terms.

(left(x+4 ight)left(2x–3 ight))

Answer

(left(x+4 ight)left(2x-3 ight))

[/hidden-answer]

The video that follows provides another example of factoring by grouping.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=118

In the next example, we will have a GCF that is negative. It is important to pay attention to what happens to the resulting binomial when the GCF is negative.

Example

Factor (3x^{2}+3x–2x–2).
[reveal-answer q=”744005″]Show Solution[/reveal-answer]
[hidden-answer a=”744005″]Group terms into pairs.

(left(3x^{2}+3x ight)+left(-2x-2 ight))

Factor the common factor 3x out of first group.

(3xleft(x+1 ight)+left(-2x-2 ight))

Factor the common factor (−2) is factored out.

(3xleft(x+1 ight)-2left(x+1 ight))

Factor out the common factor, (left(x+1 ight)), from both terms.

(left(x+1 ight)left(3x-2 ight))

Answer

(left(x+1 ight)left(3x-2 ight))

[/hidden-answer]

In the following video we present another example of factoring by grouping when one of the GCF is negative.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=118

Sometimes, you will encounter polynomials that, despite your best efforts, cannot be factored into the product of two binomials.

Example

Factor (7x^{2}–21x+5x–5).
[reveal-answer q=”262926″]Show Solution[/reveal-answer]
[hidden-answer a=”262926″]Group terms into pairs.

(left(7x^{2}–21x ight)+left(5x–5 ight))

Factor the common factor 7x out of the first group.

(7xleft(x-3 ight)+left(5x-5 ight))

Factor the common factor 5 out of the second group.

(7xleft(x-3 ight)+5left(x-1 ight))

The two groups (5left(x–1 ight)) do not have any common factors, so this polynomial cannot be factored any further.

(7xleft(x–3 ight)+5left(x–1 ight))

Answer

Cannot be factored

[/hidden-answer]

In the example above, each pair can be factored, but then there is no common factor between the pairs!

Factor Trinomials Part I

In the last section we introduced the technique of factoring by grouping as a means to be able to factor a trinomial. Now we will actually get to the work of starting with a three term polynomial, and rewriting it as a four term polynomial so it can be factored.

We will start with factoring trinomials of the form (x^2) term.

Remember that when (left(x+5 ight)), are multiplied, the result is a four term polynomial and then it is simplified into a trinomial:

(left(x+2 ight)left(x+5 ight)=x^2+5x+2x+10=x^2+7x+10)

Factoring is the reverse of multiplying, so let’s go in reverse and factor the trinomial (x^{2}), (7x=5x+2x) then we can use the grouping technique:

((x^{2}+5x)+(2x+10))

Factor each pair: (egin{array}{l}xunderbrace{left(x+5 ight)}+2underbrace{left(x+5 ight)} ext{common binomial factor}end{array})

Then pull out the common binomial factor: (left(x+5 ight)left(x+2 ight))

What would have happened if we had rewritten (6x+x)?

((x^{2}+6x)+(x+10))

Factor each pair: (xleft(x+6 ight)+1left(x+10 ight))

Then we don’t have a common factor of (left(x+5 ight)) like we did before. There is a method to the madness of choosing how to rewrite the middle terms so that you will end up with a common binomial factor.

Method to the Madness

The following is a summary of the method, then we will show some examples of how to use it.

Factoring Trinomials in the form (x^{2}+bx+c)

To factor a trinomial in the form (x^{2}+bx+c), find two integers, r and s, whose product is c and whose sum is b.

(egin{array}{l}rcdot{s}=c ext{ and } +s=bend{array})

Rewrite the trinomial as (left(x+r ight)) and (left(x+s ight)).

For example, to factor (x^{2}+7x+10), you are looking for two numbers whose sum is 7 (the coefficient of the middle term) and whose product is 10 (the last term).

Look at factor pairs of 10:1 and 10, 2, and 5. Do either of these pairs have a sum of 7? Yes, 2 and 5. So you can rewrite (2x+5x), and continue factoring as in the example above. Note that you can also rewrite (5x+2x). Both will work.

Let’s factor the trinomial (x^{2}+5x+6). In this polynomial, the b part of the middle term is 5 and the c term is 6. A chart will help us organize possibilities. On the left, list all possible factors of the c term, 6; on the right you’ll find the sums.

Factors whose product is 6Sum of the factors
(1+6=7)
(2+3=5)

There are only two possible factor combinations, 1 and 6, and 2 and 3. You can see that (2x+3x=5x), giving us the correct middle term.

Example

Factor (x^{2}+5x+6).

[reveal-answer q=”141663″]Show Solution[/reveal-answer]
[hidden-answer a=”141663″]Use values from the chart above. Replace (2x+3x).

(x^{2}+2x+3x+6)

Group the pairs of terms.

(left(x^{2}+2x ight)+left(3x+6 ight))

Factor x out of the first pair of terms

(xleft(x+2 ight)+left(3x+6 ight))

Factor 3 out of the second pair of terms.

(xleft(x+2 ight)+3left(x+2 ight))

Factor out (left(x+2 ight)).

(left(x+2 ight)left(x+3 ight))

Answer

(left(x+2 ight)left(x+3 ight))

[/hidden-answer]

Note that if you wrote (x^{2}+3x+2x+6) and grouped the pairs as (xleft(x+3 ight)+2left(x+3 ight)), and factored out (left(x+3 ight)left(x+2 ight)). Since multiplication is commutative, the order of the factors does not matter. So this answer is correct as well; they are equivalent answers.

In the following video, we present another example of how to use grouping to factor a quadratic polynomial.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=118

Finally, let’s take a look at the trinomial (−12). So look at all of the combinations of factors whose product is (−12). Then see which of these combinations will give you the correct middle term, where b is 1.

Factors whose product is (1cdot−12=−12)(2cdot−6=−12)(3cdot−4=−12)(4cdot−3=−12)(6cdot−2=−12)(12cdot−1=−12)(12+−1=11)

There is only one combination where the product is (r=4), and (s=−3). Let’s use these to factor our original trinomial.

Example

Factor (x^{2}+x–12).

[reveal-answer q=”205737″]Show Solution[/reveal-answer]
[hidden-answer a=”205737″]Rewrite the trinomial using the values from the chart above. Use values (s=−3).

(x^{2}+4x+−3x–12)

Group pairs of terms.

(left(x^{2}+4x ight)+left(−3x–12 ight))

Factor x out of the first group.

(xleft(x+4 ight)+left(-3x-12 ight))

Factor −3 out of the second group.

(xleft(x+4 ight)–3left(x+4 ight))

Factor out (left(x+4 ight)).

(left(x+4 ight)left(x-3 ight))

Answer

(left(x+4 ight)left(x-3 ight))

[/hidden-answer]

In the above example, you could also rewrite (x^{2}– 3x+4x–12) first. Then factor (left(x–3 ight)) getting (left(x–3 ight)left(x+4 ight)). Since multiplication is commutative, this is the same answer.

Factoring Tips

Factoring trinomials is a matter of practice and patience. Sometimes, the appropriate number combinations will just pop out and seem so obvious! Other times, despite trying many possibilities, the correct combinations are hard to find. And, there are times when the trinomial cannot be factored.

While there is no foolproof way to find the right combination on the first guess, there are some tips that can ease the way.

Tips for Finding Values that Work

When factoring a trinomial in the form (x^{2}+bx+c), consider the following tips.

Look at the c term first.

  • If the c term is a positive number, then the factors of c will both be positive or both be negative. In other words, r and s will have the same sign.
  • If the c term is a negative number, then one factor of c will be positive, and one factor of c will be negative. Either r or s will be negative, but not both.

Look at the b term second.

  • If the c term is positive and the b term is positive, then both r and s are positive.
  • If the c term is positive and the b term is negative, then both r and s are negative.
  • If the c term is negative and the b term is positive, then the factor that is positive will have the greater absolute value. That is, if (|r|>|s|), then r is positive and s is negative.
  • If the c term is negative and the b term is negative, then the factor that is negative will have the greater absolute value. That is, if (|r|>|s|), then r is negative and s is positive.

After you have factored a number of trinomials in the form (x^{2}+bx+c), you may notice that the numbers you identify for r and s end up being included in the factored form of the trinomial. Have a look at the following chart, which reviews the three problems you have seen so far.

Trinomial(x^{2}+5x+6)(r=+5,s=+2)(r=+4,s=–3)
Factored form(left(x+2 ight)left(x+3 ight))((x+4)(x–3))

The Shortcut

Notice that in each of the examples above, the r and s values are repeated in the factored form of the trinomial. So what does this mean? It means that in trinomials of the form (x^{2}) is 1), if you can identify the correct r and s values, you can effectively skip the grouping steps and go right to the factored form. For those of you that like shortcuts, let’s look at some examples where we use this idea.

Shortcut This Way

In the next two examples, we will show how you can skip the step of factoring by grouping and move directly to the factored form of a product of two binomials with the r and s values that you find. The idea is that you can build factors for a trinomial in this form: (x^2+bx+c) by finding r and s, then placing them in two binomial factors like this:

(left(x+r ight)left(x+s ight) ext{ OR }left(x+s ight)left(x+r ight))

Example

Factor: (y^2+6y-27)
[reveal-answer q=”601131″]Show Solution[/reveal-answer]
[hidden-answer a=”601131″]

Find r and s:

Factors whose product is -27Sum of the factors
(1-27=-26)
(3-9=-6)
(-3+9=6)

Instead of rewriting the middle term, we will use the values of r and s that give the product and sum that we need.

In this case:

(egin{array}{l}r=-3s=9end{array})

It helps to start by writing two empty sets of parentheses:

(left(,,,,,,,,,,,, ight)left(,,,,,,,,,,,, ight))

The squared term is y, so we will place a y in each set of parentheses:

(left(y,,,,,,,, ight)left(y,,,,,,,, ight))

Now we can fill in the rest of each binomial with the values we found for r and s.

(left(y-3 ight)left(y+9 ight))

Note how we kept the sign on each of the values. The nice thing about factoring is you can check your work. Multiply the binomials together to see if you did it correctly.

(egin{array}{l}left(y-3 ight)left(y+9 ight)=y^2+9y-3y-27=y^2+6y-27end{array})

Answer

(left(y-3 ight)left(y+9 ight))

[/hidden-answer]

We will show one more example so you can gain more experience.

Example

Factor: (-m^2+16m-48)
[reveal-answer q=”402116″]Show Solution[/reveal-answer]
[hidden-answer a=”402116″]

There is a negative in front of the squared term, so we will factor out a negative one from the whole trinomial first. Remember, this boils down to changing the sign of all the terms:

(-m^2+16m-48=-1left(m^2-16m+48 ight))

Now we can factor (left(m^2-16m+48 ight)) by finding r and s. Note that b is negative, and c is positive so we are probably looking for two negative numbers:

Factors whose product is 48Sum of the factors
(-1-48=-49)
(-2-12=-14)
(-3-16=-19)
(-4-12=-16)

There are more factors whose product is 48, but we have found the ones that sum to -16, so we can stop.

(egin{array}{l}r=-4s=-12end{array})

Now we can fill in each binomial with the values we found for r and s, make sure to use the correct variable!

(left(m-4 ight)left(m-12 ight))

We are not done yet, remember that we factored out a negative sign in the first step. We need to remember to include that.

(-1left(m-4 ight)left(m-12 ight))

Answer

(-m^2+16m-48=-1left(m-4 ight)left(m-12 ight))

[/hidden-answer]

In the following video, we present two more examples of factoring a trinomial using the shortcut presented here.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=118

Factor Trinomials Part II

The next goal is for you to be comfortable with recognizing where to place negative signs, and whether a trinomial can even be factored. Additionally, we will explore one special case to look out for at the end of this page.

Not all trinomials look like (x^{2}) term is 1. In these cases, your first step should be to look for common factors for the three terms.

TrinomialFactor out Common FactorFactored
(2(x^{2}+5x+6))(−5a^{2}−15a−10)(−5left(a+2 ight)left(a+1 ight))
(cleft(c^{2}–8c+15 ight))(y^{4}–9y^{3}–10y^{2})(y^{2}left(y–10 ight)left(y+1 ight))

Notice that once you have identified and pulled out the common factor, you can factor the remaining trinomial as usual. This process is shown below.

Example

Factor (3x^{3}–3x^{2}–90x).

[reveal-answer q=”298928″]Show Solution[/reveal-answer]
[hidden-answer a=”298928″]Since 3 is a common factor for the three terms, factor out the 3.

(3left(x^{3}–x^{2}–30x ight))

x is also a common factor, so factor out x.

(3xleft(x^{2}–x–30 ight))

Now you can factor the trinomial (−30) and whose sum is (−1).

The pair of factors is (5). So replace (−6x+5x).

(3xleft(x^{2}–6x+5x–30 ight))

Use grouping to consider the terms in pairs.

(3xleft[left(x^{2}–6x ight)+left(5x–30 ight) ight])

Factor x out of the first group and factor 5 out of the second group.

(3xleft[left(xleft(x–6 ight) ight)+5left(x–6 ight) ight])

Then factor out (x–6).

(3xleft(x–6 ight)left(x+5 ight))

Answer

(3xleft(x–6 ight)left(x+5 ight))

[/hidden-answer]

The following video contains two more examples of factoring a quadratic trinomial where the first step is to factor out a GCF. We use the shortcut method instead of factoring by grouping.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=118

The general form of trinomials with a leading coefficient of a is (x^{2}) term, instead of an (ax^{2}) term.

However, if the coefficients of all three terms of a trinomial don’t have a common factor, then you will need to factor the trinomial with a coefficient of something other than 1.

Factoring Trinomials in the form (ax^{2}+bx+c)

To factor a trinomial in the form (ax^{2}+bx+c), find two integers, r and s, whose sum is b and whose product is ac.

(egin{array}{l}rcdot{s}=acdot{c} +s=bend{array})

Rewrite the trinomial as (ax^{2}+rx+sx+c) and then use grouping and the distributive property to factor the polynomial.

This is almost the same as factoring trinomials in the form (a=1). Now you are looking for two factors whose product is (acdot{c}), and whose sum is b.

Let’s see how this strategy works by factoring (6z^{2}+11z+4).

In this trinomial, (b=11), and (b=11) and whose product is (acdot{c}=6cdot4=24). You can make a chart to organize the possible factor combinations. (Notice that this chart only has positive numbers. Since ac is positive and b is positive, you can be certain that the two factors you’re looking for are also positive numbers.)

Factors whose product is 24Sum of the factors
(1+24=25)
(2+12=14)
(3+8=11)
(4+6=10)

There is only one combination where the product is 24 and the sum is 11, and that is when (s=8). Let’s use these values to factor the original trinomial.

Example

Factor (11z), as (3z + 8z) (from the chart above.)

(6z^{2}+3z+8z+4)

Group pairs. Use grouping to consider the terms in pairs.

(left(6z^{2}+3z ight)+left(8z+4 ight))

Factor 3z out of the first group and 4 out of the second group.

(3zleft(2z+1 ight)+4left(2z+1 ight))

Factor out (left(2z+1 ight)).

(left(2z+1 ight)left(3z+4 ight))

Answer

(left(2z+1 ight)left(3z+4 ight))

[/hidden-answer]

In the following video, we present another example of factoring a trinomial using grouping. In this example, the middle term, b, is negative. Note how having a negative middle term and a positive c term influence the options for r and s when factoring.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=118

Before going any further, it is worth mentioning that not all trinomials can be factored using integer pairs. Take the trinomial (b=35) and whose product is (acdot{c}=2cdot7=14)? There are none! This type of trinomial, which cannot be factored using integers, is called a prime trinomial.

In some situations, a is negative, as in (−1) as the first step in factoring, as doing so will change the sign of (ax^{2}) from negative to positive, making the remaining trinomial easier to factor.

Example

Factor (−1) out of the trinomial. Notice that the signs of all three terms have changed.

(−1left(4h^{2}–11h–3 ight))

To factor the trinomial, you need to figure out how to rewrite (rs=4cdot−3=−12), and the sum of (rs=−11).

(r+s=−11)
(−12+1=−11)
(−6+2=−4)
(−4+3=−1)

Rewrite the middle term (−12h+1h).

(−1left(4h^{2}–12h+1h–3 ight))

Group terms.

(−1left[left(4h^{2}–12h ight)+left(1h–3 ight) ight])

Factor out 4h from the first pair. The second group cannot be factored further, but you can write it as (+1left(h–3 ight)=left(h–3 ight)). This helps with factoring in the next step.

(−1left[4hleft(h–3 ight)+1left(h–3 ight) ight])

Factor out a common factor of (left(h–3 ight)left(4h+1 ight)); the (+1left(h–3 ight)) in the previous step.

(−1left[left(h–3 ight)left(4h+1 ight) ight])

Answer

(−1left(h–3 ight)left(4h+1 ight))

[/hidden-answer]

Note that the answer above can also be written as (left(h–3 ight)left(−4h–1 ight)) if you multiply (−1) times one of the other factors.

In the following video we present another example of factoring a trinomial in the form (-ax^2+bx+c) using the grouping technique.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=118

Difference of Squares

We would be remiss if we failed to introduce one more type of polynomial that can be factored. This polynomial can be factored into two binomials but has only two terms. Let’s start from the product of two binomials to see the pattern.

Given the product of two binomials: (left(x-2 ight)left(x+2 ight)), if we multiply them together, we lose the middle term that we are used to seeing as a result.

Multiply:

(egin{array}{l}left(x-2 ight)left(x+2 ight) ext{}=x^2-2x+2x-2^2 ext{}=x^2-2^2 ext{}=x^2-4end{array})

The polynomial (x^2-4) is called a difference of squares because teach term can be written as something squared. A difference of squares will always factor in the following way:

Factor a Difference of Squares

Given (left(a+b ight)left(a-b ight))

Let’s factor (x^{2}+0x–4). This is similar in format to the trinomials we have been factoring so far, so let’s use the same method.

Find the factors of (acdot{c}) whose sum is b, in this case, 0:

Factors of (1cdot-4=−4)(2cdot−2=−4)(-1cdot4=−4)(-1+4=3)

2, and -2 have a sum of 0. You can use these to factor (x^{2}–4).

Example

Factor (0x) as (−2x+2x).

(egin{array}{l}x^{2}+0x-4x^{2}-2x+2x-4end{array})

Group pairs.

(left(x^{2}–2x ight)+left(2x–4 ight))

Factor x out of the first group. Factor 2 out of the second group.

(xleft(x–2 ight)+2left(x–2 ight))

Factor out (left(x–2 ight)).

(left(x–2 ight)left(x+2 ight))

Answer

(left(3x–2 ight)left(3x+2 ight))

[/hidden-answer]

Since order doesn’t matter with multiplication, the answer can also be written as (left(x+2 ight)left(x–2 ight)).

You can check the answer by multiplying (left(x–2 ight)left(x+2 ight)=x^{2}+2x–2x–4=x^{2}–4).

The following video show two more examples of factoring a difference of squares.

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=118

Summary

When a trinomial is in the form of (ax^{2}+bx+c), find two integers, r and s, whose sum is b and whose product is ac. Then rewrite the trinomial as (ax^{2}+rx+sx+c) and use grouping and the distributive property to factor the polynomial.

When (−1) out of the whole trinomial before continuing.

A difference of squares (left(a+b ight)left(a-b ight)).

Summary

Trinomials in the form (x^{2}+rx+sx+c) and then use grouping and the distributive property to factor the polynomial.


6.4: More Factoring Methods

This textbook is primarily about quantitative research, in part because most studies conducted in psychology are quantitative in nature . Quantitative researchers typically start with a focused research question or hypothesis, collect a small amount of data from a large number of individuals, describe the resulting data using statistical techniques, and draw general conclusions about some large population. Although this method is by far the most common approach to conducting empirical research in psychology, there is an important alternative called qualitative research. Qualitative research originated in the disciplines of anthropology and sociology but is now used to study psychological topics as well. Qualitative researchers generally begin with a less focused research question, collect large amounts of relatively “unfiltered” data from a relatively small number of individuals, and describe their data using nonstatistical techniques. They are usually less concerned with drawing general conclusions about human behavior than with understanding in detail the experience of their research participants.

Consider, for example, a study by researcher Per Lindqvist and his colleagues, who wanted to learn how the families of teenage suicide victims cope with their loss (Lindqvist, Johansson, & Karlsson, 2008) [1] . They did not have a specific research question or hypothesis, such as, What percentage of family members join suicide support groups? Instead, they wanted to understand the variety of reactions that families had, with a focus on what it is like from their perspectives. To address this question, they interviewed the families of 10 teenage suicide victims in their homes in rural Sweden. The interviews were relatively unstructured, beginning with a general request for the families to talk about the victim and ending with an invitation to talk about anything else that they wanted to tell the interviewer. One of the most important themes that emerged from these interviews was that even as life returned to “normal,” the families continued to struggle with the question of why their loved one committed suicide. This struggle appeared to be especially difficult for families in which the suicide was most unexpected.


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Operationalization

Once a theoretical construct is defined, exactly how do we measure it? Operationalization refers to the process of developing indicators or items for measuring these constructs. For instance, if an unobservable theoretical construct such as socioeconomic status is defined as the level of family income, it can be operationalized using an indicator that asks respondents the question: what is your annual family income? Given the high level of subjectivity and imprecision inherent in social science constructs, we tend to measure most of those constructs (except a few demographic constructs such as age, gender, education, and income) using multiple indicators. This process allows us to examine the closeness amongst these indicators as an assessment of their accuracy (reliability).

Indicators operate at the empirical level, in contrast to constructs, which are conceptualized at the theoretical level. The combination of indicators at the empirical level representing a given construct is called a variable . As noted in a previous chapter, variables may be independent, dependent, mediating, or moderating, depending on how they are employed in a research study. Also each indicator may have several attributes (or levels) and each attribute represent a value . For instance, a “gender” variable may have two attributes: male or female. Likewise, a customer satisfaction scale may be constructed to represent five attributes: “strongly dissatisfied”, “somewhat dissatisfied”, “neutral”, “somewhat satisfied” and “strongly satisfied”. Values of attributes may be quantitative (numeric) or qualitative (non-numeric). Quantitative data can be analyzed using quantitative data analysis techniques, such as regression or structural equation modeling, while qualitative data require qualitative data analysis techniques, such as coding. Note that many variables in social science research are qualitative, even when represented in a quantitative manner. For instance, we can create a customer satisfaction indicator with five attributes: strongly dissatisfied, somewhat dissatisfied, neutral, somewhat satisfied, and strongly satisfied, and assign numbers 1 through 5 respectively for these five attributes, so that we can use sophisticated statistical tools for quantitative data analysis. However, note that the numbers are only labels associated with respondents’ personal evaluation of their own satisfaction, and the underlying variable (satisfaction) is still qualitative even though we represented it in a quantitative manner.

Indicators may be reflective or formative. A reflective indicator is a measure that “reflects” an underlying construct. For example, if religiosity is defined as a construct that measures how religious a person is, then attending religious services may be a reflective indicator of religiosity. A formative indicator is a measure that “forms” or contributes to an underlying construct. Such indicators may represent different dimensions of the construct of interest. For instance, if religiosity is defined as composing of a belief dimension, a devotional dimension, and a ritual dimension, then indicators chosen to measure each of these different dimensions will be considered formative indicators. Unidimensional constructs are measured using reflective indicators (even though multiple reflective indicators may be used for measuring abstruse constructs such as self-esteem), while multidimensional constructs are measured as a formative combination of the multiple dimensions, even though each of the underlying dimensions may be measured using one or more reflective indicators.


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Highest Common Factor



Examples, examples, and videos to help GCSE Maths students learn how to find the highest common factor (HCF).

What is Highest Common Factor (HCF)?
The Highest Common Factor of two or more numbers is the largest number that can divide the numbers without any remainder. The highest common factor is also called the Greatest Common Factor (GCF)

The following diagrams show the methods that can be used to find the Highest Common Factor. Scroll down the page for more examples and solutions on how to find the Highest Common Factor.

How to find the HCF?
There are various methods to find the HCF. Here are some of them:
1. List out all the factors of each number and select the largest factor that is common to all the lists.
2. Use prime factors or factor trees.
3. Use division (ladder) method.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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Healthcare Additive Manufacturing Market Size Worth $6.4 Billion By 2028: Grand View Research, Inc.

SAN FRANCISCO , Feb. 3, 2021 /PRNewswire/ -- The global healthcare additive manufacturing market size is expected to be valued at USD 6.4 billion by 2028 and is anticipated to grow at a CAGR of 21.8% from 2021 to 2028, according to a new report by Grand View Research, Inc. Easy development of customized products, reduction in manufacturing cost due to technological advancements, and possibilities of using various material for printing are propelling the market growth.

Key suggestions from the report:

  • Laser sintering accounted for the largest market share of 31.0% in 2020 as it uses a wide variety of materials to make high-quality, complex geometries and can produce several pieces at one time
  • The polymers material segment accounted for the largest revenue share of more than 50% in 2020, as polymer-based AM has been used for decades in creating medical instruments as well as prosthetic limbs & related accessories
  • North America led the market in 2020 and accounted for the largest share of more than 35% due to the presence of several additive manufacturing companies with a robust distribution network
  • Asia Pacific is projected to be the fastest-growing regional market over the forecast period due to significant demand for dental 3D printing as a result of an increasing number of people undergoing tooth replacement surgeries.

Read 114 page research report with ToC on "Healthcare Additive Manufacturing Market Size, Share & Trends Analysis Report By Technology (Laser Sintering, Deposition Modeling), By Application (Medical Implants, Wearable Devices), By Material, And Segment Forecasts, 2021 - 2028" at: https://www.grandviewresearch.com/industry-analysis/healthcare-additive-manufacturing-market

The growing success of additive manufacturing is due to its benefits over conventional manufacturing methods. Some of the benefits include the application of advanced technology use of a wide range of materials like metal, plastics, and polymers flexibility in design build speed dimensional accuracy and its ability to produce complex parts/geometry, such as cooling channels and honeycomb structure. Production of implants and prosthetics is the largest application for additive manufacturing in the healthcare industry.

There is an increasing demand for orthopedic procedures, such as knee & hip replacements, due to the increasing geriatric population. Moreover, the increasing application of dental implants along with a growing need for prosthetics is another key factor expected to drive the market. North America and Europe were the leading regional markets in 2020. This growth was attributed to extensive R&D, advanced healthcare system, and the presence of leading 3D printing machinery manufacturers & material suppliers in these regions.

Additive manufacturing is playing a significant role in the fight against COVID-19 by compensating for the shortage of medical supplies by speeding up the manufacturing process. This will drive the growth of the market over the forecast period. However, stringent regulatory approvals associated with medical devices might hinder the growth.

Grand View Research has segmented the global healthcare additive manufacturing market on the basis of technology, application, material, and region:

  • Healthcare Additive Manufacturing Technology Outlook (Revenue, USD Million, 2016 - 2028)
    • Stereolithography
    • Deposition Modeling
    • Electron Beam Melting
    • Laser Sintering
    • Jetting Technology
    • Laminated Object Manufacturing
    • Others
    • Medical Implants
    • Prosthetics
    • Wearable Devices
    • Tissue Engineering
    • Others
    • Metals & Alloys
    • Polymers
    • Biological Cells
    • Others
    • North America
      • U.S.
      • Canada
      • U.K.
      • Germany
      • Spain
      • France
      • Italy
      • China
      • India
      • Japan
      • Australia
      • South Korea
      • Brazil
      • Mexico
      • Argentina
      • Colombia
      • South Africa
      • Saudi Arabia
      • UAE

      List of Key Players of Healthcare Additive Manufacturing Market

      • GE Additive (General Electric)
      • 3D Systems, Inc.
      • EnvisionTEC GmbH
      • RegenHU
      • Allevi, Inc.
      • EOS GmbH (Electro Optical Systems)
      • Materialise N.V.
      • Stratasys Ltd.
      • Nanoscribe GmbH
      • GPI Prototype and Manufacturing Services, LLC.

      Find more research reports on Medical Devices Industry, by Grand View Research:

        – The global 3D printing market size was valued at USD 11.58 billion in 2019 and is expected to expand at a CAGR exceeding 14% from 2020 to 2027. Globally, 1.42 million units of 3D printers were shipped in 2018 and this number is expected to reach 8.04 million units by 2027. – The global restorative dentistry market size was valued at USD 14.59 billion in 2016 and is expected to grow at a CAGR of 6.7% during the forecast period. – The global prosthetics and orthotics market size was estimated at USD 5.9 billion in 2019 and is expected to exhibit a CAGR of 4.6% during the forecast period.

      Gain access to Grand View Compass, our BI enabled intuitive market research database of 10,000+ reports


      6.4: More Factoring Methods

      Here are the steps required for Solving Quadratics by Factoring:

      Step 1: Write the equation in the correct form. To be in the correct form, you must remove all parentheses from each side of the equation by distributing, combine all like terms, and finally set the equation equal to zero with the terms written in descending order.
      Step 2: Use a factoring strategies to factor the problem.
      Step 3: Use the Zero Product Property and set each factor containing a variable equal to zero.
      Step 4: Solve each factor that was set equal to zero by getting the x on one side and the answer on the other side.

      Example 1 &ndash Solve: x 2 + 16 = 10x

      Example 2 &ndash Solve: 18x 2 – 3x = 6

      Example 3 &ndash Solve: 50x 2 = 72

      Example 4 &ndash Solve: x(2x – 1) = 3

      Example 5 &ndash Solve: (x + 3)(x – 5) = 𔃅


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      Use of Present Value Annuity Factor Formula

      The present value annuity factor is used for simplifying the process of calculating the present value of an annuity. A table is used to find the present value per dollar of cash flows based on the number of periods and rate per period. Once the value per dollar of cash flows is found, the actual periodic cash flows can be multiplied by the per dollar amount to find the present value of the annuity.

      For example, an individual is wanting to calculate the present value of a series of $500 annual payments for 5 years based on a 5% rate. By looking at a present value annuity factor table, the annuity factor for 5 years and 5% rate is 4.3295. This is the present value per dollar received per year for 5 years at 5%. Therefore, $500 can then be multiplied by 4.3295 to get a present value of $2164.75.


      Watch the video: Solving Polynomials via Factoring (October 2021).