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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Factoring is an essential skill for success in algebra and higher level mathematics courses. Therefore, we have taken great care in developing the student's understanding of the factorization process. The technique is consistently illustrated by displaying an empty set of parentheses and describing the thought process used to discover the terms that are to be placed inside the parentheses.The factoring scheme for special products is presented with both verbal and symbolic descriptions, since not all students can interpret symbolic descriptions alone. Two techniques, the standard "trial and error" method, and the "collect and discard" method (a method similar to the "ac" method), are presented for factoring trinomials with leading coefficients different from 1. This module contains the exercise supplement for the chapter "Factoring Polynomials".

Exercise supplement

Finding the factors of a monomial ( [link] )

For the following problems, the first quantity represents the product and the second quantity represents a factor. Find the other factor.

44 a 2 b 2 c , 11 b 2

4 a 2 c

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50 m 3 n 5 p 4 q , 10 m 3 q

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51 ( a + 1 ) 2 ( b + 3 ) 4 , 3 ( a + 1 )

17 ( a + 1 ) ( b + 3 ) 4

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26 ( x + 2 y ) 3 ( x y ) 2 , 13 ( x y )

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8 x 5 y 4 ( x + y ) 4 ( x + 3 y ) 3 , 2 x ( x + y ) ( x + 3 y )

4 x 4 y 4 ( x + y ) 3 ( x + 3 y ) 2

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( 6 a 5 b ) 10 ( 7 a b ) 8 ( a + 3 b ) 7 , ( 6 a 5 b ) 7 ( 7 a b ) 7 ( a + 3 b ) 7

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12 x n + 6 y 2 n 5 , 3 x n + 1 y n + 3

4 x 5 y n 8

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400 a 3 n + 10 b n 6 c 4 n + 7 , 20 a 2 n + 8 c 2 n 1

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16 x 32 , 16

( x 2 )

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24 a 2 6 a , 6 a

4 a 1

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88 x 4 33 x 3 + 44 x 2 + 55 x , 11 x

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9 y 3 27 y 2 + 36 y , 3 y

3 y 2 + 9 y 12

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4 m 6 16 m 4 + 16 m 2 , 4 m

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5 x 4 y 3 + 10 x 3 y 2 15 x 2 y 2 , 5 x 2 y 2

x 2 y 2 x + 3

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21 a 5 b 6 c 4 ( a + 2 ) 3 + 35 a 5 b c 5 ( a + 2 ) 4 , 7 a 4 b ( a + 2 ) 2

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x 2 y c 2 , 1

x + 2 y + c 2

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Factoring a monomial from a polynomial ( [link] ) - the greatest common factor ( [link] )

For the following problems, factor the polynomials.

3 y 2 + 27 y

3 y ( y + 9 )

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21 ( x + 5 ) + 9

3 ( 7 x + 38 )

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m a 3 m

m ( a 3 1 )

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r 2 ( r + 1 ) 3 3 r ( r + 1 ) 2 + r + 1

( r + 1 ) [ r 2 ( r + 1 ) 2 3 r ( r + 1 ) + 1 ]

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( 10 - 3 x ) ( 2 + x ) + 3 ( 10 3 x ) ( 7 + x )

( 10 3 x ) ( 23 + 4 x )

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Factoring by grouping ( [link] )

For the following problems, use the grouping method to factor the polynomials. Some may not be factorable.

x y + 4 x 3 y 12

( x 3 ) ( y + 4 )

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2 a b 8 b 3 a b 12 a

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a 2 7 a + a b 7 b

( a + b ) ( a 7 )

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r 2 + r s r s

( r 1 ) ( r + s )

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8 a 2 b c + 20 a 2 b c + 10 a 3 b 3 c + 25 a 3 b 3

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a ( a + 6 ) ( a + 6 ) + a ( a 4 ) ( a 4 )

2 ( a + 1 ) ( a 1 )

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a ( 2 x + 7 ) 4 ( 2 x + 7 ) + a ( x 10 ) 4 ( x 10 )

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Factoring two special products ( [link] ) - factoring trinomials with leading coefficient other than 1 ( [link] )

For the following problems, factor the polynomials, if possible.

m 2 36

( m + 6 ) ( m 6 )

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a 2 + 8 a + 16

( a + 4 ) 2

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m 2 + m + 1

not factorable

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a 2 + 9 a + 20

( a + 5 ) ( a + 4 )

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x 2 + 14 x + 40

( x + 10 ) ( x + 4 )

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n 2 14 n + 49

( n 7 ) 2

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a 2 9 a + 20

( a 5 ) ( a 4 )

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4 a 2 9 a 9

( 4 a + 3 ) ( a 3 )

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42 a 2 + 5 a 2

( 6 a 1 ) ( 7 a + 2 )

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56 m 2 + 26 m + 6

2 ( 28 m 2 + 13 m + 3 )

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4 x 2 + 4 x y 3 y 2

( 2 x + 3 y ) ( 2 x y )

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2 x 2 + 6 x 20

2 ( x 2 ) ( x + 5 )

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x 3 + 3 x 2 4 x

x ( x + 4 ) ( x 1 )

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3 y 4 27 y 3 + 24 y 2

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15 a 2 b 2 a b 2 b

b ( 15 a 2 b a 2 )

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18 a 2 6 a + 1 2

( 6 a 1 ) ( 3 a 1 2 )

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4 x 2 12 x y + 9 y 2

( 2 x 3 y ) 2

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r 6 s 8 + 6 r 3 s 4 p 2 q 6 + 9 p 4 q 12

( r 3 s 4 + 3 p 2 q 6 ) 2

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a 4 2 a 2 b 15 b 2

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81 a 8 b 12 c 10 25 x 20 y 18

( 9 a 4 b 6 c 5 + 5 x 10 y 9 ) ( 9 a 4 b 6 c 5 5 x 10 y 9 )

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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