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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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Questions & Answers

find the equation of the tangent to the curve y=2x³-x²+3x+1 at the points x=1 and x=3
Esther Reply
derivative of logarithms function
Iqra Reply
how to solve this question
sidra
ex 2.1 question no 11
khansa
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khansa
question please
Rasul
ex 2.1 question no. 11
khansa
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khansa
Find the derivative of g(x)=−3.
Abdullah Reply
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need to learn polynomial
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Tlou
evaluate the following computation (x³-8/x-2)
Murtala Reply
teach me how to solve the first law of calculus.
Uncle Reply
teach me also how to solve the first law of calculus
Bilson
what is differentiation
Ibrahim Reply
only god knows😂
abdulkadir
f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
Naufal Reply
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
diron
pweding paturo nsa calculus?
jimmy
how to use fundamental theorem to solve exponential
JULIA Reply
find the bounded area of the parabola y^2=4x and y=16x
Omar Reply
what is absolute value means?
Geo Reply
Chicken nuggets
Hugh
🐔
MM
🐔🦃 nuggets
MM
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
find integration of loge x
Game Reply
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
Godwin Reply
how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
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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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