# Factor Polynomials by Common Factor

Questions With Detailed Solutions

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

## What is factorization by common factor?It is a factorization method based on the law of distributivity which is a(b + c) = a · b + a · c and 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.
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 expanding the factored form to check if you get the original polynomial Example: check that 3 x^{ 2} - x = x(3x - 1)
Expand x(3x - 1) using the law of distributivity.
x(3x - 1) = (x)(3x) +(x)(-1) = 3x^{2} - x , which is correct.
## More ExamplesFind a common factor and use the method of distributivity in reverse to factor the polynomials completely.a) 9 x - 6b) x^{ 2} - xc) 3 x + 12 x yd) 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 examplesa) 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 ·3The greatest common factor is 3 and is factored out. Hence 9 x - 6 = 3 (3 x - 2)b) The prime factorization of x and ^{ 2}x is needed to find the greatest common factor in x.
^{ 2} - xx^{ 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 and ^{ 2} y 4 x y are needed to find the greatest common factor in ^{ 2}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 · yThe greatest common factor is 2 · 2 · x = 4 x. Hence
16 x) = 4 x (4 x^{ 3} + 8 x^{ 2} y + 4 x y^{ 2} = 4 x ( 2 · 2 · x · x + 2 · x · y + y · y^{ 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 and ^{ 4}x and factor it out.
^{ 2}2 x^{ 4} + x^{ 2} = 2 · x · x · x · x + x · x = x^{ 2}(2 x^{ 2} + 1)The complete factoring of 2 x is written as follows:
^{ 4}(x + 5) + x^{ 2}(x + 5)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 polynomialsDetailed 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. |

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