FACTORING POLYNOMIALS

Methods to Factor Polynomials

Any Polynomial—Look for the Greatest Common Factor :

xy - xz  =  x(y - z)

Example : 

6a2b + 10ab2  =  2ab(3a + 5b)

Binomials—Look for a Difference of Two Squares :

x2 - y2  =  (x + y)(x - y)

Example : 

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 : 

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 : 

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 : 

2y3 + 4y2 + y + 2 : 

=  (2y3 + 4y2) + (y + 2)

=  2y2(y + 2) + 1(y + 2)

=  (y + 2)(2y2 + 1)

Note : 

If none of the factoring methods work, the polynomial is unfactorable.

Remember : 

For a polynomial of the form ax2 + bx + c, if there are no integers whose sum is b and whose product is ac, then the polynomial is unfactorable.

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 : 

2a(a2 + 4)

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

2a(a2 + 4) is completely factored. 

Example 2 : 

(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 3 :

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)=  -2a(b2 - 8b + 16)

=  -2ab2 + 16ab - 32a 

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 :

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.

Factors of 4

-1 and -4

Sum

-5 

The factors needed are -1 and -4.

=  3m2(m - 1)(m - 4)

Example 6 :

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 7 :

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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