**Factoring Quadratic Polynomials in Two Steps**

We will now work in the opposite direction—factoring. First we will factor *quadratic polynomials*, expressions of the form *ax ^{2}* +

*bx*+

*c*(where

*a*is not 0). For example

*x*

^{2}+ 5

*x*+ 6 is factored as (

*x*+ 2)(

*x*+ 3). Quadratic polynomials whose first factors are

*x*

^{2}are the easiest to factor. Their factorization always begins as (

*x*±__)(

*x*±__). This forces the first factor to be

*x*

^{2}when the

**FOIL**method is used. All you need to do is fill in the two blanks and decide when to use plus and minus signs. All quadratic polynomials factor though some do not factor “nicely.” We will only concern ourselves with “nicely” factorable polynomials in this chapter.

**Step 1: Determine the Signs**

If the second sign is minus, then the signs in the factors will be different (one plus and one minus). If the second sign is plus then both of the signs will be the same. In this case, if the first sign in the trinomial is a plus sign, both signs in the factors will be plus; and if the first sign in the trinomial is a minus sign, both signs in the factors will be minus.

**Examples**

x^{2} – 4*x* – 5 = ( *x* – __)( *x* +__) or ( *x* + __)( *x* – __)

x^{2} + *x* – 12 = ( *x* +__)( *x* – __) or ( *x* – __)( *x* + __)

x^{2} – 6*x* + 8 = ( *x* – __)( *x* – __)

x^{2} + 4*x* + 3 = ( *x* + __)( *x* + __)

Find practice problems and solutions at Factoring Quadratic Polynomials Practice Problems - Set 1.

**Step 2: Determine the Constants**

Once the signs are determined all that remains is to fill in the two blanks. Look at all of the pairs of factors of the constant term. These pairs will be the candidates for the blanks. For example, if the constant term is 12, you will need to consider 1 and 12, 2 and 6, and 3 and 4. If both signs in the factors are the same, these will be the only ones you need to try. If the signs are different, you will need to reverse the order: 1 and 12 as well as 12 and 1; 2 and 6 as well as 6 and 2; 3 and 4 as well as 4 and 3. Try the FOIL method on these pairs. (Not every trinomial can be factored in this way.)

**Examples**

**Example 1:**

*x*^{2} + *x* – 12

Factors to check: (*x* + 1)(*x* – 12), (*x* – 1)(*x* + 12), (*x* + 2)(*x* – 6), (*x* – 2)(*x* + 6), (*x* – 4)(*x* + 3), and (*x* + 4)(*x* – 3).

(*x* + 1)(*x* – 12) = *x*^{2} – 11*x* – 12

(*x* – 1)(*x* + 12) = *x*^{2} + 11*x* – 12

(*x* + 2)(*x* − 6) = *x*^{2} – 4*x* – 12

(*x* – 2)(*x* + 6) = *x*^{2} + 4*x* – 12

(*x *– 4)(*x* + 3) = *x*^{2} – *x* – 12

(*x* + 4)(*x* – 3) = *x*^{2} + *x* – 12 (This works.)

**Example 2: **

*x*^{2} – 2*x* – 15

Factors to check: (*x* + 15)(*x* – 1), (*x* – 15)(*x* + 1), (*x* + 5)(*x* – 3), and (*x* – 5)(*x* + 3) (works).

*x*^{2} – 11*x* + 18

Factors to check: (*x* – 1)(*x* – 18), (*x* – 3)(*x* – 6), and (*x* – 2) (*x* – 9) (works).

*x*^{2} + 8*x* + 7

Factors to check: (*x* + 1)(*x* + 7) (works).

Find practice problems and solutions at Factoring Quadratic Polynomials Practice Problems - Set 2.

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