# 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 :

=  -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) Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

1. Click on the HTML link code below.

Featured Categories

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test 