Download Calculus 30 A2 – Factoring

Calculus 30 A2 – Factoring
A. Factor each of the following completely over the set of rational numbers.
1. 27a3b4 18a2b11
2. 3xy 2 z 3  15 x 3 yz 2  30 x 2 y 3 z
x2  81
4x2  25h2
125  5x2
32t 4  2
25
7. r 2 
49
3.
4.
5.
6.
8.
9.
 3m  1  9
2
2
5x  y    2 y  z 
2
10. f 3  27
11. v3  1000
12. 125a3  64b3
14. 10x2  29x  21
13. x2  7 x  18
15. tan3 x 1
16.
x
2
 x  4
2
17. x 2 log 2 x  4 log 2 x
B. Factor each of the following as a difference of squares over the set of real numbers.
1. x2 10
2. 4x2  48
3. a  8
4. sin x  2sin3 x
C. Factor each of the following as a sum or difference of cubes over the set of real numbers.
1. x3 10
2. t 3  24
3. k  8
D. Factor each of the following completely over the set of rationals by initially regarding each expression
as a difference of squares.
1. x6 1
2. 64a6  81b6
E. Factor each of the following completely over the set of rationals by initially regarding each expression as
a difference of cubes.
1. x6 1
2. 64a6  81b6
F. By examining the discriminant, determine which of the following trinomials can be factored over the set
of rationals.
1. x2  4x  16
2. x2  3x  9
3. 6x2  5x  56
4. 8x2  5x  48
© Saskatchewan Learning
G. Factor each expression. Begin by eliminating all fractional coefficients.
1.
4 2
x 3
3
2.
1 2 1
1
x  x
6
36
3
1 2
x  20
5
3 3 4 1 6
x y  xy
4.
4
3
1 2 1
5.
x  x2
4
2
3.
H. Factor each of the following expressions so that the second factor contains no negative exponents and no
fractional coefficients.
1. 2x3  12x2
2. 21a4  7a3
3. 54 x3 y 6  45 x 3 y 8
4. 16b1 2  12b1 2
5. 35a2 3b3 4 10a1 3b5 4
1 2 3 1 5 3  7 3  6 5
g t  g t
6.
2
8
3 3 2 3 2 1 1 2 1 2
x y
 x y
7.
20
15
I. Factor each of the following using the result that
x n  y n   x  y   x n 1  x n  2 y  x n 3 y 2  ...  x 2 y n 3  xy n  2  y n 1 
1. x 7  y 7
2. x6 1
3. x5  32
J. Factor each of the following polynomials over the rationals by grouping:
1. 2x3  5x2 18x  45 Hint: x2  2 x  5  9  2 x  5
2. 2x5  2x4 12x3  5x2  5x  30
K. Factor each of the following polynomials over the rationals by using the factor theorem together with
synthetic division.
1. x3  7 x  6 Hint: p 1  0 , thus x  1 is a factor.
2. x4  x3 11x2  9x  18
3. 6x4  x3  8x2  x  2
© Saskatchewan Learning
Calculus 30 A2 – Factoring
Answer Key
A
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
9a2b4(3a - 2b7)
-3xyz(yz2 - 5x2z + 10xy2)
(x + 9)(x - 9)
(2x - 5h)(2x + 5h)
5(5 - x)(5 + x)
2(2t - 1)(2t + 1)(4t2 + 1)
(r - 5/7)(r + 5/7)
(3m + 4)(3m - 2)
(5x + 3y - z)(5x - y + z)
(f - 3)(f2 + 3f + 9)
(v + 10)(v2 - 10v + 100)
(5a - 4b)(25a2 + 20ab + 16b2)
(x - 9)(x + 2)
(5x - 3)(2x + 7)
(tanx - 1)(tan2x – tanx + 1)
(x + 2)(x - 1)(x2 + x + 2)
log2x(x + 2)(x - 2)
B.
1.) ( x  10)( x  10) hint let a = x and b = 10
2.) (2 x  4 3)(2 x  4 3) hint let a = 2x and b = 4 3
3.) ( a  2 2)( a  2 2) hint let a=
a and b= 2 2
4.) sin x(1  2 sin x)(1  2 sin x)
C
3
3
3
3
1.) ( x  10)( x  10 x  100) : Hint let a = x and b= 10
2
3
3
3
2.) (t  2 3t )(t  2 3t  4 9) : Hint let a = t and b =
2
3.) ( 3
23 3
3
k  2)( 3 k 2  2 3 k  4) : Hint let a = k and b = 2
D
1.) ( x
3
2.) (8a
 1)( x3  1) : Hint let a = x3 and b = 1
3
 9b3 )(8a3  9b3 ) : Hint let a = 8a3 and b = 9b3
E
1.) ( x  1)( x  1)( x 4  x 2  1) : Hint let a = x2 and b = 1
3
3
2.) (4a  81b )(16a  4 81a b  b
2
2
4
2 2
43
6561) : Hint let a = 4a2 and b = b2 3 81
F
b2  4ac = 48 can not be factored over real number system
2.) 25 , can not be factored over real number system
1.)
3.) 37, can be factored over real number system
© Saskatchewan Learning
4.)
1561 , can’t be factored over real number system
G.
1.
4 2
1
1
x  3   4 x2  9   2 x  3 2 x  3
3
3
3
1 2 1
1 6
1
12
x  x   x2  x 
6
36
3 36
36
36
1
  2 x  33x  4
36
1 2
1
1
3. x  20   x2 100   x 10 x  10
5
5
5
3 3 4 1 6 9 3 4 4 6 xy 4
xy 4
x y  xy  x y  xy 
9 x2  4 y 2  
4.
 3x  2 y  3x  2 y 

4
3
12
12
12
12
1 2 1
1
2
8 1
1
5.
x  x  2  x2  x    x2  2x  8   x  4)( x  2
4
2
4
4
4 4
4
2.
H.

1. 2 x3  12 x 2  2 x 3 1  6 x5
4
2. 21a  7a
3 6
3. 54x y
3

4
 7a (3  a)
 45x3 y8  9x3 y8 (6x6 y2  5)
4. 16b1 2 12b1 2  4b1 2 (4  3b)
5. 35a2 3b3 4 10a1 3b5 4  5a1 3b3 4 (7a  2b2 )
4 2 3 1 5 3 7 3 6 5 1 7 3 6 5
g t  g t  g t (4g 3t  3)
8
8
8
9 3 2 3 2 4 1 2 1 2 1 1 2 3 2 2
1
7.
x y  x y  x y (9x  4 y 2 )  x1 2 y 3 2 (3x  2 y)(3x  2 y)
60
60
60
60
6.
I.
1. x7  y7  ( x  y)( x6  x5 y  x4 y 2  x3 y3  x2 y 4  xy5  y6 )
2. x6 1  ( x 1)( x5  x4  x3  x2  x 1)
3. x5  32  ( x  2)( x4  2x3  4x2  8x 16)
J.
1. x2  2x  5  9  2x  5  ( x2  9)(2 x  5)  ( x  3)( x  3)(2x  5)
2. 2x3 ( x2  x  6)  5( x2  x  6)  (2x3  5)( x2  x  6)
K. Factor each of the following polynomials over the rationals by using the factor theorem together with
synthetic division.
1. (x-2)(x-1)(x+3)
2. (x-3)(x-2)(x+1)(x+3)
3. (x-1)(x+1)(2x-1)(3x+2)
© Saskatchewan Learning