FACTORING POLYNOMIALS

Methods to Factor Polynomials

Any Polynomial—Look for the Greatest Common Factor :

xy - xz  =  x(y - z)

Example 1 : 

6a²b + 10ab²  =  2ab(3a + 5b)

Binomials—Look for a Difference of Two Squares :

x² - y²  =  (x + y)(x - y)

Example 2 : 

a² - 9b²  =  (a + 3b)(a - 3b)

Trinomials—Look for Perfect-Square Trinomials :

x² + 2xy + y²  =  (x + y)²

x² - 2xy + y²  =  (x - y)²

Examples 3 : 

a² + 4a + 4  =  (a + 2)²

a² - 2a + 1  =  (a - 1)²

Other Factorable Trinomials :

x² + bx + c  =  (x + _ ) (x + _ )

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

Examples 4 : 

y² + 3y + 2  =  (y + 1)(y + 2)

6y² + 7y + 2  =  (2y + 1)(3y + 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 5 : 

2y³ + 4y² + y + 2 : 

=  (2y³ + 4y²) + (y + 2)

=  2y²(y + 2) + 1(y + 2)

=  (y + 2)(2y² + 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 6 : 

2a(a² + 4)

Neither 2a nor a² + 4 can be factored further.

2a(a² + 4) is completely factored. 

Example 7 : 

(2a + 6)(a + 5)

2a + 6 can be further factored.

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

=  2(a + 3)(a + 5)

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

Factoring by GCF and Recognizing Patterns

Example 8 :

Factor -2ab² + 16ab - 32a completely. Check your answer.

=  -2ab² + 16ab - 32a

Factor out the GCF.

=  -2a(b² - 8b + 16)

b² + 8b + 16 is a perfect square trinomial of the form

x² + 2xy + y²

x = b and y = 4.

=  -2a(b - 4)²

Check : 

-2a(b - 4)²  =  -2a(b² - 8b + 16)

=  -2ab² + 16ab - 32a ✓

Factoring by Multiple Methods

Factor each polynomial completely.

Example 9 :

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 10 :

3m⁴ - 15m³ + 12m²

Factor out the GCF.

=  3m²(m² - 5m + 4)

There is no pattern.

=  3m²(m + _ )(m + _ )

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

The factors needed are -1 and -4.

=  3m²(m - 1)(m - 4)

Example 11 :

4y³ + 18y² + 20y

Factor out the GCF.

=  2y(2y² + 9y + 10)

There is no pattern.

=  2y( _ y + _ )( _ y + _ )

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

=  2y(y + 2)(2y + 5)

Example 12 :

n⁵ - n

Factor out the GCF.

=  n(n⁴ - 1)

n⁴ - 1 is a difference of two squares.

=  n(n² + 1)(n² - 1)

n² - 1 is a difference of two squares.

=  n(n² + 1)(n + 1)(n - 1)

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