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Proficiency Exam

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: 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 a proficiency exam for the chapter "Factoring Polynomials".

Proficiency Exam

Exercise 1

((Reference)) The product is 27 a 3 +9 a 2 +9a 27 a 3 +9 a 2 +9a and a factor is 3a 3a . Find the other factor.

Solution

9 a 2 +3a+3 9 a 2 +3a+3

Exercise 2

((Reference)) The product is 15⁢xn+5⁢y3⁢n−2 15 x n 5 y 3 n 2 . Find the other factor.

Solution

5 x n y 2n−3 5 x n y 2n−3

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

Exercise 3

((Reference)) −14 x 2 y 4 b−28 x 2 y 3 b-42 x 2 y 2 −14 x 2 y 4 b−28 x 2 y 3 b-42 x 2 y 2

Solution

−14 x 2 y 2 ( y 2 b+2yb+3 ) −14 x 2 y 2 ( y 2 b+2yb+3 )

Exercise 4

((Reference)) ( y+2 ) a +( y+2 ) c ( y 2 ) a ( y 2 ) c

Solution

( a+c )( y+2 ) ( a+c )( y+2 )

Exercise 5

((Reference)) 6 x 2 y 2 z + 5 x 2 y 3 −12 x y z−10x y 2 6 x 2 y 2 z 5 x 2 y 3 12 x y z 10x y 2

Solution

xy( xy−2 )( 6z+5y ) xy( xy−2 )( 6z+5y )

Exercise 6

((Reference)) 4 a 2 − 16 c 2 4 a 2 16 c 2

Solution

4( a+2c )( a−2c ) 4( a+2c )( a−2c )

Exercise 7

((Reference)) m 4 − n 4 m 4 n 4

Solution

( m 2 + n 2 )( m+n )( m−n ) ( m 2 + n 2 )( m+n )( m−n )

Exercise 8

((Reference)) b 2 +8b+16 b 2 +8b+16

Solution

( b+4 ) 2 ( b+4 ) 2

Exercise 9

((Reference)) 9 y 2 −30 y+25 9 y 2 30 y 25

Solution

( 3y−5 ) 2 ( 3y−5 ) 2

Exercise 10

((Reference)) x 2 +5x−15 x 2 5x 15

Solution

not factorable

Exercise 11

((Reference)) x 2 −x−30 x 2 x 30

Solution

( x−6 )( x+5 ) ( x−6 )( x+5 )

Exercise 12

((Reference)) 4 x 6 − 36 x 4 + 80 x 2 4 x 6 36 x 4 80 x 2

Solution

4 x 2 ( x 2 −5 )( x+2 )( x−2 ) 4 x 2 ( x 2 −5 )( x+2 )( x−2 )

Exercise 13

((Reference)) 9 x 2 +25x−6 9 x 2 +25x−6

Solution

( 9x−2 )( x+3 ) ( 9x−2 )( x+3 )

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