Introduction
When factoring a polynomial, you want to change the sum and difference of many terms into the product of different polynomials.
For example, when asked to factor the number 12, you might write 2*2*3, because these are the smallest numbers that would multiply to make 12.
It is important that we be able to do this with polynomials as well. Breaking them down into then product of smaller pieces (factors) allows us to simplify problems.
When factoring a trinomial (a polynomial made up of 3 terms), you should first look for a GCF.
Secondly, if the “x2” term does not have a number in front of it (called a coefficient), then we should consider the easier factoring method.
If there is a coefficient in front of the “x2” term (our “a” value), we need to factor it into two binomials.
To determine what they would look like, we do what’s called a “diamond problem.” (The “diamond” looks more like an “X.”) First we need to multiply the “a” value by our “c” value from the polynomial and put this in the top of the diamond. Then the “b” coefficient goes into the bottom of the diamond.
Then, we need to solve this diamond problem by coming up with two numbers on the left and right of the diamond that multiply to the top number and add to the bottom number.
*Hint* It may be helpful to list out all the factors (numbers that multiply to a given number) to help
you find the pair that adds to the “b” number.
Examples
Given 3x2+ 17x + 10
First, we multiply our “a” value of 3 times our “c” value of 10 to get 30.
In our diamond problem (“x” form to the left), we put the 30 at the top and the 17 at the bottom. We are now looking for factors of 30 that add to 17. factors of 30: 1 & 30, 2 & 15, 3 & 10, 5 & 6
we can see that the factors 2 & 15 add to 17.
We then use the 2 and 15 to break up the middle term of the trinomial 3x2 + 17x + 10 into separate terms like this:
We then factor this using the grouping method
x(3x + 2) + 5( 3x + 2)
(3x + 2)(x + 5)
Check your answer by FOIL-ing or re-multiplying these two binomials together to verify that you get the original polynomial back again, thereby confirming you factored it correctly.
Here’s a link to a video walking you through this method:
Practice Problems: Now you try!
1. 6x2 + 13x + 6
2. 7x2 + 15x + 2
3. 3x2– 2x – 8
4. 2x2 + 7x – 30
5. 5x2 – 14x + 8
6. 12x2– 7x + 1
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