Factoring Simple Trinomials

What is a Trinomial?

A trinomial is a polynomial with 3 terms. It is written in standard form

(ax^2+ bx + c), where "a", "b", and "c" are just numbers. A simple trinomial always has an a-value of 1. This makes it easy to factor the trinomial because all you will have to find are two numbers that will not only multiply to equal the constant term c, but also add up to equal b.

Let’s do an example…

Factor the following: x^2 + 7x + 6

To factor the above equation you need to find the factors of 6 that add up to 7.

6 can be written as the product of 1 and 6

Also, 1 + 6 = 7

So the factors are 1 and 6.

From multiplying polynomials you know the equation is formed when you multiply two binomials of the form "(x + r) (x +s)". Since you have already found the factors, all you have to do is replace the r and the s with those numbers.

Therefore…x^2 + 7x + 6= (x + 2)(x + 3)

Now you try…

Factor the Following: x^2 + 7x + 12

Answer…

(x + 3)(x + 4)

Let’s do another example with a negative sign…

Factor the following: x^2 + 5x - 14

Answer…

? x ? = -14

? + ? = 5

Criteria Needed…

Factors with opposite signs since we have a negative product
A negative smaller factor since we have a positive middle term.

Since-14 can be written as the product of -2 and 7, and since -2+ 7 = 5,

x^2 + 5x – 14

=(x - 2)(x + 7)

Let’s try an example where the sum is negative and the product is positive…

Factor the following: x^2 - 6x + 8

Answer…

? x ? = 8

? + ? = -6

Criteria Needed…

Both factors need to be negative since we have a negative sum.
This would work because we have a positive product (Remember negative x negative = positive).

So 8 can be written as the product of -2 and -4, and since -2 - 4 = - 6, the factors are -2 and - 4.

x^2 - 6x + 8

=(x - 2)(x – 4)

Let’s try one last equation, which has two negative signs…

Factor the following: x^2 -7x - 18

Answer…

? x ? = -18

? + ? = -7

Criteria Needed…

Factors with opposite signs.
A negative larger factor since the middle term is negative.

So -18 can be written as the product of 2 and -9, and since 2 - 9 = - 7, the factors are -7 and - 18.

x^2 -7x -18

=(x - 9)(x + 2)

**Always remember to look at the sign in front of the product and in front of the sum**

Here is a table to help you determine the signs in front of each factor:

Also if the sum is positive, then the larger factor will be positive and if the sum is negative, then the larger factor will also be negative.

Extra Practice…

A Video for Extra Help…