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2.5: Factoring the GCF - Mathematics


The distributive property of multiplication over addition/subtraction can be reversed.

(a(bpm c)=abpm ac) (right side equals left side) implies (abpm ac=a(bpm c)) (left side equals right side).

Factoring is the art of taking a sum (addition of terms) or difference (subtraction of terms) into a product (multiplication of factors).

Example (PageIndex{1})

Factor (15x+20y).

Solution

[egin{array}{rcl lll} 15x+20y&=&3cdot 5x+4cdot 5y &=&5(3x+4y) end{array}]


GCF example:

The first step is to find all divisors of each number. For instance, let us find the gcf(2,5,15).

  • The factors of 2 (all the whole numbers that can divide the number without a remainder) are 1 and 2
  • The factors of 5 are 1 and 5
  • The factors of 15 are 1, 3, 5 and 15.

The second step is to analyze which are the common divisors. It is not difficult to see that the 'Greatest Common Factor' or 'Divisor' for 2, 5 and 15 is 1. The GCF is the largest common positive integer that divides all the numbers (2,5,15) without a remainder.

  • Greatest common divisor (gcd)
  • Highest common factor (hcf)
  • Greatest common measure (gcm), or
  • Highest common divisor

Greatest Common Factor Calculator

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There are, basically, three methods of solving Quadratic Equations by Factoring:

Use the Sum-Product Method in Solving Quadratic Equations by Factorizing

This method is mainly used by students who find it challenging to use the guessing method, (or the trial and error method). Unlike the trial and error method, the Product Sum Method is generally easier to apply since it identifies an equation that cannot be factored.

This method takes various forms, i.e:

Case 1

Simply follow these steps when solving an equation using the product sum method:

Step One: Find two integers whose product is C.

Step Two: Give the integers any characters of your choice, for example, M and N.

Step Three: Make one factor ( X + M ) and the other ( X + N).

Illustration 1

Find the value of x by factorization.

Find two integers whose product is 15. The table below shows the numbers.

Below are the pairs of the numbers

You can then select the pair that has the sum of 16 and product 55.

Therefore, the factors are X² + 16X +55 = (X +5)(X +11) = 0,

Illustration 2

Find the value of X in X² -16X +60 =0

Identify a duo of integers whose product is 60 the pairs are listed below.

You can have a table with different values for different pairs.

You should then select a pair that has sum -16.

Case 2

If the equation AX²+BX + C =0 and A≠1 you only need a little extra effort to find the factors using the product sum method.

Here are the steps to follow:

  1. Identify two integers whose product is AC and sum is B.
  2. You can name the integers M and N.
  3. Rewrite the function as a four term expression as below AX² + MX + NX + C.
  4. Use grouping by pair to factor out the Greatest Common Factor (GCF) in the two terms to get a common parenthesis.

To illustrate this case, let's consider the following examples.

Example 1

Find the value of X given that 2X²+ X -10=0

Solution

Find two integers whose product AC= (2)×(-10)=-20. You should then draw a table on your working paper to come up with several pairs.

You can now select the pair that has the sum of B = 1. This pair is - 4 and 5.

Rewrite the expression as 2X²- 4X + 5x - 10 = 0

Taking out GCF, we get 2X(X-2)5(x-2).

Now we have the common parenthesis, which is X-2.

Therefore,(2X +5)(X-2)= 0 where X = 2 or X = -5/2

Example 2

Calculate the value of X given that 3X²+X-2 =0

Solution

Find two integers whose product is AC= 3 ×-2=-6

Next, list the pairs in a tabular form.

Select the pair that has the sum of B=1. This pair is 3 and -2

You should then rewrite the function as 3X²+3X-2X-2= 0.

Now the common parenthesis is X+1

How to Solve a Quadratic Equation by Factoring Using Grouping Method

This method involves arranging the terms into smaller groupings with common factors. Use the factoring by grouping method if you can't find the common factor for all the terms.

Further, by taking two terms at the same time, you can get something to divide the terms.

Use this easy procedure in solving the equation by factorizing using the grouping method.

Suppose you are given a general equation AX² +BX + C

  1. Find the product of AC.
  2. Think of two numbers, say Q and P such that QP= AC and Q+P= B
  3. Rewrite the expression as AX² + QX +PX +C
  4. Group the expression into two pairs that have a common factor and simplify like this:

Depending on your selection of P and Q, you will factor out a constant on the second parenthesis, remaining with two identical expressions as shown in the example below:

Example 1

Find the value of X given 5X² + 11X +2= 0

Solution

First find the product AC

Then think of two factors of 10 that can add up to 11

Next, write 11X in the product of 10 and 1.

You should now group the pairs into two.

After grouping, take out the common factor.

Example 2

Compute the value of X given that X²+2X-24=0

Solution

First, find the product AC = 1×-24=-24

Think of two factors, such that their product is -24 and their sum is 2.

Let the factors be -4 and +6

Next, write +2X in the form -4X and 6X

Therefore, the expression becomes X²- 4X +6X-24=0

Pair the equation into 2 terms, thus:

Next, take out the common factor.

Now, X-4 becomes the common parenthesis.

Solving Quadratic Equations by Factorizing Using the Special Product Method

The Special Product Method requires special cases that can be factored quicker. You can do this using two special quadratics:

Case 1: Perfect Square Expression

  1. Check whether the first and the last term are perfect squares
  2. Check whether the middle term is 2 times the product of the roots of the other terms.

Illustration

Identify the value of X given that X²+ 14X +49=0

Solution

The first term is X² and the last term is 49 both X² and 49 are perfect squares whose roots are X and 7 respectively.

The middle term, 14X is two times the roots of the other terms.

Case 2: Difference of Two Squares Expression

Special Product Method is used here since:

  1. There are no common factors
  2. The typical middle term is missing
  3. The terms present are perfect squares and being subtracted.

Example 1

Identify the value of X given that X²-16=0

Solution

Note that in the quadratic equation above:

  1. The middle term is missing.
  2. The terms present² and 16 are perfect squares and are being subtracted.

Example 2

Find the value of X given that X2-64=0

The terms X² and 64 are perfect squares and they are subtracted.

Verdict

Quadratics are considered to be among the most challenging concepts in Mathematics.

Regardless, getting the correct methods and learning how to apply the concepts can make teaching and learning Mathematics fun!

Consequently, knowledge of quadratics is important in everyday life.

For instance, quadratics is used in the determination of profits or even in formulating the speed and velocity of an object.

Further, quadratics have been applied to athletic endeavours like shotput and javelin. Take time to learn the various methods to solve quadratic equations using the Factorizing method and you'll come to love it!


More factoring & prime factors worksheets

Explore all of our factoring worksheets, from factoring numbers under 50 to prime factor trees, GCFs and LCMs.

K5 Learning offers free worksheets, flashcards and inexpensive workbooks for kids in kindergarten to grade 5. We help your children build good study habits and excel in school.

K5 Learning offers free worksheets, flashcards and inexpensive workbooks for kids in kindergarten to grade 5. We help your children build good study habits and excel in school.


Greatest Common Factor Calculator (GCF or GCD)

On this page is a greatest common factor calculator, often abbreviated as GCF. This term goes by many names – it is also known as the greatest common divisor (GCD) and highest common factor (HCF).

Enter a set of numbers and the GCF tool will return the greatest common factor. While the tool will accept negatives and decimals, you should use positive integers .

Looking for a similar tool? Try one of these instead:


2.5: Factoring the GCF - Mathematics

Question from Peter, a student:

I need to know the differences between GCF and LCD.

The least common denominator LCD of two fractions is the least common multiple LCM of the denominators. So I am going to compare the greatest common factor GCF and LCM of two positive integers. The words themselves tell you want they are

greatest common factor
least common multiple

Lets look at the GCF first (sometimes called the greatest common divisor GCD). Consider the numbers 36 and 60. I am looking for common factors of these two numbers, that is positive integers that divide both of them. 1 divides them both, so does 2 and 3 and 6 but what is the greatest positive integer that divides them both? I can answer this if I write the prime factorization of both 36 and 60.

So 2 2 divides them both, 3 divides them both but not 3 2 . 5 divided 60 but not 36. No other prime number divides them. Thus the greatest positive integer that divided them both is 2 2 × 3 = 12. Thus

What about the least common multiple LCM of say 12 and 9? This time we want a multiple of 12 and 9. Certainly 12 × 9 = 108 is a multiple of them both but we want the least common multiple so is there a multiple which is less than 108? Again look at the prime factorizations.


Greatest Common Factors

Sometimes we need to be able to find the greatest commonfactor of a set of numbers. The greatest common, or GCF, is thelargest number that will divide evenly into each of the numbersin a set.

Find the GCF for the set of numbers: 10, 12, 20

The largest number that will go into each of these numbers is2.

Find the GCF for the set of numbers: 6, 18, 36

The largest number that will go into each of these numbers is6.

Find the GCF for the set of numbers: 4, 8, 10

The largest number that will go into each of these numbers is2.

Find the GCF for the set of numbers: 8a 2 b, 18a 2 b 2 c

The first thing we do is find the GCF for the coefficients -just like we've been doing. The largest number that will go intoeach of the coefficients is 2.

Since we have variables, we have to find their GCF also. For avariable to be included in the GCF, each term must have thevariable. If the variable is in each term, we take the lowestexponent of the variable and include it in the GCF.

In this case, both terms have a and both terms have b . Wewill include a 2 because that is the lowest power of a. We will include b because that is the lowest power of b .

Find the GCF for the set of numbers: 3x 2 y, 12x 4 y 2 , 9x 2 y

The first thing we do is find the GCF for the coefficients.The largest number that will go into each of the coefficients is3.

Since we have variables, we have to find their GCF also. For avariable to be included in the GCF, each term must have thevariable. If the variable is in each term, we take the lowestexponent of the variable and include it in the GCF.

In this case, both terms have x and both terms have y . Wewill include x 2 because that is the lowest power of x. We will include y because that is the lowest power of y .


How can GCF be used in real life?

We use greatest common factors all the time with fractions, and as fractions are used a lot in everyday life, this makes GCF very useful!

By finding the GCF of the denominator and numerator, you can then successfully simplify a fraction or ratio.

E.g. We can simplify #30/45# by knowing that its HCF is #15# .
Then we divide both parts by the HCF to simplify.
#(30/15)/(45/15) = 2/3#

It also works for ratios, where you can simplify each side using HCF to find out a #1:X# ratio. This can be useful if you are using a ratio for a recipe or order as you can use one piece of information to find out the right ratio for any combination.

So, to put this into a situation, say you know that for every 5 people at a party, you need 15 sandwiches. The HCF of these two numbers is 5, so for each person you need:

3 sandwiches.
Now, if 16 people come to your party, you know you have to make #16xx3 = 48# sandwiches.

A final example is with recipes.
This is a very useful time for maths to get involved!

Here is a recipe for a 10 cupcakes and their ratios related to the serving size :
100g flour = 10 people:100g = 1:10
80g sugar = 10 people:80g = 1:8
50g butter = 10 people:50g = 1:5
2 eggs = 10 people:2 eggs = 1:0.2 eggs

So, if we want to give cakes to all our friends, and need 25 cupcakes (what a popular mathematician!) then you can just multiply out this ratio.

Flour = 1:10 = 25:250
80g sugar = 1:8 = 1:200
50g butter = 1:5 = 25:125
2 eggs = 1:0.2 eggs = 25:5


Watch the video: mathΑΙΝΩ με παράδειγμα #5 - Μαθηματική λογική 1 (October 2021).