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factoring trinomials


A trinomial is a polynomial made up of three terms. Factoring trinomials is the inverse of multiplying two binomials



Instead of multiplying two binomials to get a trinomial, you will write the trinomial as a product of two binomials

The general form of a trinomial is ax2 + bx + c

Your goal in factoring trinomials is to make ax2 + bx + c equal to (? + ?) * (? + ?)

When a = 1, the trinomial becomes x2 + bx + c and it is easier to factor. Therefore, I will start by showing you how to factor when a = 1.

Example #1:

Factor x2 + 5x + 6

x2 + 5x + 6 will look like (x + ?) * (x + ?)

We are 100 % sure that the first term for each binomial must be x because x * x = x2

factoring-image


Now, how do we get the second term for each binomial?

We also know for sure that ? * ? or the product of the second term for each binomial is equal to 6

factoring-image


Finally, we know that x * ? and ? * x must give the second term, which is 5x when added

factoring-image


Thus,when factoring trinomials, the trick is to look for factors of 6(last term), that will add up to 5(coefficient of second term)

-6 is equal to:

6 × 1

-6 × - 1

2 × 3

-2 × -3

The only pair of factors that will add up to 5 is 2 and 3 because 2 + 3 = 5

Just replace the two question mark by 2 and 3 and you are done

Therefore, x2 + 5x + 6 = (x + 3) * (x + 2)

Notice that (x + 3) * (x + 2) = also equal to (x + 2) * (x + 3) since multiplication is commutative

The final step is to check your answer by multiplying the two binomials

x * x = x2

x * 2 = 2x

3 * x = 3x

3 * 2 = 6

Since 2x + 3x = 5x, putting it all together, we get:

x2 + 5x + 6

Example #2:

Factor x2 −5x + 6

It is almost the same equation as before with the exception that the coefficient of the second term is -5 instead of 5

Follow all steps outlined above. The only difference is that you will be looking for factors of 6 that will add up to -5 instead of 5.

-3 and -2 will do the job

So, x2 −5x + 6 = (x + -3) * (x + -2)

Final example

Factor x2 −x −20

First, notice that x2 −x −20 = x2 −1x −20 because 1*x = x

x2 −x −20 = (x + ?) * (x + ?)

Find factors of -20 that will equal to -1

-20 is equal to

-20 * 1

20 * -1

10 * -2

-10 * 2

4 * -5

-4 * 5

Since 4 + -5 = -1, we have found what we need. x2 −x −20 = (x + 4) * (x + -5)





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