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Factoring trinomials is the inverse of multiplying two binomials. Instead of multiplying two binomials to get a trinomial, you write the trinomial as a product of two binomials.
x² + 5x + 6 = (x + 2)(x + 3)
Work through a new trinomial one step at a time, or test yourself in quiz mode.
Find two numbers that multiply to . There are pairs to find, including negative ones.
Score: 0 / 0
The general form of a trinomial is ax² + bx + c. Your goal when factoring trinomials is to write ax² + bx + c as (? + ?)(? + ?).
When a = 1, the trinomial becomes x² + bx + c and it is easier to factor. This lesson only shows you how to factor when a = 1.
x² + 5x + 6 will look like (x + ?)(x + ?)
Be 100% sure that the first term of each binomial must be x, because x × x = x².
How do we get the second term of each binomial? We know for sure that ? × ?, the product of the second terms, is equal to 6.
x × ? and ? × x must give the middle term, 5x, when they are added.
x × ? + ? × x = x(? + ?)
So the two numbers we do not know must also add up to 5.
The trick: look for factors of the last term that add up to the coefficient of the middle term.
The last term is 6 and the coefficient of the middle term is 5. It can help to list all the factor pairs of 6 with their sums.
The only pair that adds up to 5 is 2 and 3. Replace the two question marks with 2 and 3, and you are done.
x² + 5x + 6 = (x + 2)(x + 3)
(x + 2)(x + 3) is also equal to (x + 3)(x + 2), since multiplication is commutative.
Since 2x + 3x = 5x, putting it all together gives x² + 5x + 6. ✓
It is almost the same trinomial as before, except that the coefficient of the middle term is −5 instead of 5. Follow the same steps, but look for factors of 6 that add up to −5. Each pair below shows its sum.
Since −2 and −3 do the job, x² − 5x + 6 = (x + (−2))(x + (−3)). Adding a negative number is the same as subtracting, so we write:
x² − 5x + 6 = (x − 2)(x − 3)
First, notice that x² − x − 20 = x² − 1x − 20, because 1 × x = x. So it will look like (x + ?)(x + ?), where the two numbers multiply to −20 and add up to −1. Each pair below shows its sum.
Since 4 + (−5) = −1, we have found what we need.
x² − x − 20 = (x + 4)(x − 5)
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