Factorization requires that you remember a lot of the things we’ve done previously. Please use the opportunity to revise if we mention anything you’ve forgotten. Before we look at our questions, let’s look at a few things.

First, remember that when we looked at addition and subtraction in Algebra, we said you cannot add, let’s say, 6x squared plus 9xy . This doesn’t mean there’s nothing you can do to this expression. in fact we will be able to factorize it.

Second, Note that factoring in algebra is really a form of reversal of multiplication.

In our algebra question 7, we looked at multiplying the monomial 3x  by the polynomial 2x + 3y.

We said we will multiply each term of the polynomial by the monomial to get 6x squared  + 9xy.

When factoring, you’ll be given this as your question and your work will be to get this as your answer. you are finding what multiplied to get you this values.

In order to achieve this reversal, all we have to do is bring the common values outside the parenthesis, and the uncommon values inside the parenthesis. Let’s actually look at this question

 

Question 1.

Factor  6x² + 9xy.

We said factoring requires us bringing the common values outside the parenthesis and the uncommon values inside the parenthesis.

The common value is basically the greatest common factor or GCF. Let's start with the numbers. We learned 3 ways of finding the GCF of numbers on question 20 of our GED basic math section. You can use any of the methods. I'm using the factor method.

Factors of 6 are, 1, 2, 3, and 6.

Factors of 9 are, 1, 3, and 9.

3 is the greatest number common to both numbers, so 3 is the GCF.

We bring it outside the parenthesis.

Now we divide each of the numbers by the 3 outside.

6 divided by 3 is 2.

And 9 divided by 3 is 3.

We are done with the number part.

Now we check if there are common letters. We have x common to both.

For letters, the GCF is the letter with the smallest exponent.

Here, we have x exponent 2 and x exponent 1. We know that x is the same as x exponent 1.

since exponent 1 is the smallest, x exponent 1 is the GCF so we will bring it out.

We will divide each of the x values by the X outside the parenthesis like we did with the 3.

From the Quotient Rule of exponents, we know that when you do such division, you can simply subtract the exponents.

so for the first term we have x exponent 2 minus 1, to get x exponent 1, which is simply x.

For the second term we have x exponent 1 divided by x exponent 1. 

That's x exponent 1 minus 1. This will give x exponent zero. Meaning there'll be no x term here. 

There are no more common letters. We have an uncommon y here, so we will write it here.

So we have 3x, in parenthesis 2x + 3y as our answer, like we expected.

 

Question 2.

Simplify (x² - 64) ÷ (x + 8)

A very common factorization is factoring the difference of two squares.

If you have, let’s say, x squared minus y squared, then this factorize to become x minus y times x + y . 

Please, this only works if the two values are squared and the sign between them is minus.

Let’s look at our question.

The first things you should recognize is that 64 is the same as 8 squared. so we can replace the 64 with 8 squared.

Now we see we have the difference of two squares. This will become x minus 8, times x + 8 over x + 8.

The x + 8 will cancel out since we have it in both the numerator and denominator. 

What’s left is x minus 8. Therefore our answer is x minus 8.

Question 31.

We are factoring the quadratic, x squared minus, 5x minus 6.

Remember we said that factoring is basically a reversal of multiplication. For this question we are reversing what we did when we looked at polynomial-by-polynomial multiplication. You can revise that.

Our work is to find two numbers that multiplies to get the constant. In this case the -6.

The two numbers must also add up to get the coefficient of the x, which is -5.

This takes some trial and error, and depends mainly on how good your multiplication is.

The two numbers here will be -6 and 1.

-6 times one will be -6

-6 plus 1 will be -5.

After finding these numbers, all you’ll do is put them in parenthesis with the x and you are done.

So we have x minus 6 and x + 1

Please, this method works only if there’s no number in front of the x squared. In math, we say the coefficient of the x squared is 1.