CHOOSING A FACTORING METHOD

Methods to Factor Polynomials

Any Polynomial—Look for the Greatest Common Factor :

ab - ac = a (b - c)

Example : 

6x²y + 10xy²  =  2xy(3x + 5y)

Binomials—Look for a Difference of Two Squares :

a² - b²  =  (a + b)(a - b)

Example : 

x² - 9y²  =  (x + 3y)(x - 3y)

Trinomials—Look for Perfect-Square Trinomials :

a² + 2ab + b²  =  (a + b)²

a² - 2ab + b²  =  (a - b)²

Examples : 

x² + 4x + 4  =  (x + 2)²

x² - 2x + 1  =  (x - 1)²

Other Factorable Trinomials :

x² + bx + c  =  (x + _ ) (x + _ )

ax² + bx + c  =  ( _ x + _ ) ( _ x + _ )

Examples : 

x² + 3x + 2  =  (x + 1)(x + 2)

6x² + 7x + 2  =  (2x + 1)(3x + 2)

Polynomials of Four or More Terms - Factor by grouping :

ax + bx + ay + by : 

=  x(a + b) + y(a + b) 

=  (x + y)(a + b)

Example : 

2x³ + 4x² + x + 2 : 

=  (2x³ + 4x²) + (x + 2)

=  2x²(x + 2) + 1(x + 2)

=  (x + 2)(2x² + 1)

Factoring Polynomials 

Recall that a polynomial is in its fully factored form when it is written as a product that cannot be factored further.

To factor a polynomial completely, you may need to use more than one factoring method. Use the steps below to factor a polynomial completely.

Step 1 : 

Check for a greatest common factor.

Step 2 : 

Check for a pattern that fits the difference of two squares or a perfect-square trinomial.

Step 3 : 

To factor x² + bx + c, look for two numbers whose sum is b and whose product is c.

To factor ax² + bx + c, check factors of a and factors of c in the binomial factors. The sum of the products of the outer and inner terms should be b.

Step 4 : 

Check for common factors.

Determining Whether an Expression Is Completely Factored

Tell whether each expression is completely factored. If not, factor it.

Example 1 : 

2y(y² + 4)

Neither 2y nor y² + 4 can be factored further.

2y(y² + 4) is completely factored. 

Example 2 : 

(2x + 6)(x + 5)

2x + 6 can be further factored.

Factor out 2, the GCF of 2x and 6.

=  2(x + 3)(x + 5)

2(x + 3)(x + 5) is completely factored.

Factoring by GCF and Recognizing Patterns

Example 3 :

Factor -2xy² + 16xy - 32x completely. Check your answer.

=  -2xy² + 16xy - 32x

Factor out the GCF.

=  -2x(y² - 8y + 16)

y² + 8y + 16 is a perfect square trinomial of the form

a² + 2ab + b²

a = y and b = 4.

=  -2x(y - 4)²

Check : 

-2x(y - 4)²  =  -2x(y² - 8y + 16)

=  -2xy² + 16xy - 32x ✓

Factoring by Multiple Methods

Factor each polynomial completely.

Example 4 :

2x² + 5x + 4

The GCF is 1 and there is no pattern.

=  ( _ x + _ ) ( _ x + _ )

a = 2 and c = 4; Outer + Inner = 5. 

2x² + 5x + 4 is unfactorable.

Example 5 :

3n⁴ - 15n³ + 12n²

Factor out the GCF.

=  3n²(n² - 5n + 4)

There is no pattern.

=  3n²( n + _ ) (n + _ )

b = -5 and c = 4; look for factors of 4 whose sum is -5.

The factors needed are -1 and -4.

=  3n²(n - 1)(n - 4)

Example 6 :

4x³ + 18x² + 20x

Factor out the GCF.

=  2x(2x² + 9x + 10)

There is no pattern.

=  2x( _ x + _ ) ( _ x + _ )

a = 2 and c = 10; Outer + Inner = 9

=  2x(x + 2)(2x + 5)

Example 7 :

p⁵ - p

Factor out the GCF.

=  p(p⁴ - 1)

p⁴ - 1 is a difference of two squares.

=  p(p² + 1) (p² - 1)

p² - 1 is a difference of two squares.

=  p(p² + 1) (p + 1)(p - 1)

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