When we multiply two polynomials together we can write the product in expanded form. If we were to write an expanded polynomial as the product of two (or more) polynomials, we would be putting the polynomial into factored form.

factored form -->   (x + 2)(x + 3) = x²+5x + 6   <-- expanded form

To factor means to rewrite a polynomial as a product of polynomials. In the above example (x + 2) and (x + 3) are the factors of (x²+ 5x + 6).

How do we factor a trinomial like x²+7x + 12?

Recall our general expansion:

(x + a)(x + b) = x² + (a + b)x + ab

The sum of a and b gives us the coefficient of the x term and the product of a and b gives us the constant term.

In the example x²+7x + 12 we need to find two numbers such that their sum is +7 and their product is +12.

Start by listing the factors of 12 in a chart, along with the sum of the factors.

Factors	Sum
1, 12	13
2, 6	8
3, 4	7
-3, -4	-7
-2, -6	-8
-1, -12	-13

 

We can see from the chart that the only pair of numbers that have a product or +12 and a sum of +7 are 3 and 4.

Thus we can factor as follows:

x²+ 7x + 12 = (x + 3)(x + 4)

 

 

 

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Example 1: Factor x²+ x - 6.

We need to find a pair of numbers that sum to +1 and have a product of -6.

x2+ x - 6 = (x - 2)(x + 3)

Example 2: Factor x²- 6x +9.

Sum must = -6
Product must = +9

(x²- 6x + 9) = (x - 3)(x - 3)
                     = (x - 3)²

Solve the following.

1. (x - 21)(x +6)	2. (x + 18)(x - 10)
3. (x + 7)(x - 14)	4. (x +5)(x + 19)

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