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.
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 - 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 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 ·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 polynomials.
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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