Factor Polynomials by Common Factor - Grade 11 Math Questions With Detailed Solutions

How to factor a polynomial using a common factor? Grade 11 math questions are presented along with detailed solutions. Detailed Solutions and explanations are included.

Factoring a polynomial is to write it as the product of simpler polynomials.

Example:

2 x + 4 = 2(x + 2)

3 x 2 - x = x(3x - 1)

NOTE: it is very easy to check if your factorization is correct by multiplying to see if you get the original polynomial

Example: check that 3 x 2 - x = x(3x - 1)

Expand x(3x - 1) by mulitplication

x(3x - 1) = (x)(3x) +(x)(-1) = 3x2 - x , which is correct.

What is factorization by common factor?

It is a factorization method based on the law of distributivity

a(b + c) = a b + a c


used in reverse as follows

a b + a c = a(b + c)


a is a common factor to a b and a c is therefore factored out.

Example: Find a common factor and use the method of distributivity in reverse to factor the polynomials completely.

a)
9 x - 6

b)
x 2 - x

c)
3 x + 12 x y

d)
16 x 3 + 8 x 2 y + 4 x y 2

e)
2 x 4(x + 5) + x 2(x + 5)


Solution to the above examples

a) Find any common factors in the two terms of
9 x - 6 by expressing both terms 9x and 6 in the given binomial as prime factorization. Hence

9 x - 6 = 3 3 x - 2 3

The greatest common factor is
3 and is factored out. Hence

9 x - 6 = 3 (3 x - 2)


b) The prime factorization of
x 2 and x is needed to find the greatest common factor in x 2 - x.

x 2 - x = x x - x = x x - 1 x

The greatest common factor is
x and is therefore factored out. Hence

x 2 - x = = x (x - 1)


c) The prime factorizations of
3 x and 12 x y are needed to find the greatest common factor in 3 x + 12 x y.

3 x + 12 x y = 3 x - 3 4 x y = 3 x 1 - 3 x 4 y

The greatest common factor is
3 x. Hence

3 x + 12 x y = 3 x (1 + 4 y)


d) The prime factorization of
16 x 3 , 8 x 2 y and 4 x y 2 are needed to find the greatest common factor in 16 x 3 + 8 x 2 y + 4 x y 2.

16 x 3 + 8 x 2 y + 4 x y 2 = 2 2 2 2 x x x + 2 2 2 x x y + 2 2 x y y

The greatest common factor is
2 2 x = 4 x. Hence

16 x 3 + 8 x 2 y + 4 x y 2 = 4 x ( 2 2 x x + 2 x y + y y) = 4 x (4 x 2 + 2 x y + y 2)


e) We note that
x + 5 is a common factor which can be factored out as follows:

2 x 4(x + 5) + x 2(x + 5) = (x + 5)(2 x 4 + x 2)

We now find the greatest common factor of the terms
2 x 4 and x 2 and factor it out.

2 x 4 + x 2 = 2 x x x x + x x = x 2(2 x 2 + 1)

The complete factoring of
2 x 4(x + 5) + x 2(x + 5) is written as follows:

2 x 4(x + 5) + x 2(x + 5) = x 2(x + 5)(2 x 2 + 1)

Use common factors to factor completely the following polynomials.

Detailed Solutions and explanations to these questions.

a) - 3 x + 9

b) 28 x + 2 x 2

c) 11 x y + 55 x 2 y

d) 20 x y + 35 x 2 y - 15 x y 2

e) 5 y (x + 1) + 10 y 2(x + 1) - 15 x y (x + 1)

Detailed Solutions and explanations to these questions.

More Middle School Math (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers

More High School Math (Grades 10, 11 and 12) - Free Questions and Problems With Answers

More Primary Math (Grades 4 and 5) with Free Questions and Problems With Answers

Author - e-mail

Home Page


Updated: 20 January 2017 (A Dendane)
Share
Additional Info