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Practice
Factoring Polynomials
Easy
Normal
Difficult
Factoring Polynomials: Very Difficult Problems with Solutions
By
Catalin David
Problem 1
Factor 3x
3
- x
2
y +6x
2
y - 2xy
2
+ 3xy
2
- y
3
=
(3x - 2y)(x + y)
(3x - y)(x + y)(x - y)
(3x - y)(x + y)
2
(3x - y)(x
2
+ y
2
)
Solution:
3x
3
- x
2
y + 6x
2
y - 2xy
2
+ 3xy
2
- y
3
=
x
2
(3x - y) + 2xy(3x - y) + y
2
(3x-y) =
(3x - y)(x
2
+ 2xy + y
2
)=
(3x - y)(x + y)
2
Problem 2
Factor (x + 4)
3
- 9x - 36 =
(x - 4)(x + 1)(x + 7)
(x + 6)(x + 2)(x + 9)
(x + 4)(x + 1)(x + 5)
(x + 4)(x + 1)(x + 7)
Solution:
(x + 4)
3
- 9x - 36=
(x + 4)
3
- 9(x + 4)=
(x + 4)[(x + 4)
2
- 9]=
(x + 4)(x + 4 - 3)(x + 4 + 3)=
(x + 4)(x + 1)(x + 7)
Problem 3
Factor x
2
+ 2x + 1 - y
2
=
(x + 1 - y)(x + 1 + y)
(x - 1 - y)(x + 1 + y)
(x + 1 - y)(x + 1 - y)
(2x + 1 - y)(x + 1 + y)
Solution:
x
2
+ 2x + 1 - y
2
=
(x
2
+ 2x + 1) - y
2
=
(x + 1)
2
- y
2
=
(x + 1 - y)(x + 1 + y)
Problem 4
Factor x
2
- 6x + 9 - 4y
2
=
(x - 3 - 2y)(x + 3 - 2y)
(x - 3 - y)(x - 3 + 2y)
(x + 3 + 2y)(x - 3 + 2y)
(x - 3 - 2y)(x - 3 + 2y)
Solution:
x
2
- 6x + 9 - 4y
2
=
(x
2
- 6x + 9) - 4y
2
=
(x - 3)
2
- (2y)
2
=
(x - 3 - 2y)(x - 3 + 2y)
Problem 5
Factor a
2
- b
2
- 10b - 25 =
(a + b - 5)(a + b + 5)
(a - b - 5)(a + b + 5)
-(a + b + 5)(a + b - 5)
(a - b - 5)(a + b + 1)
Solution:
a
2
- b
2
- 10b - 25 =
a
2
- (b
2
+ 10b + 25) =
a
2
- (b + 5)
2
=
[a - (b + 5)][a + (b + 5)] =
(a - b - 5)(a + b + 5)
Problem 6
Factor a
2
- b
2
+ 12b - 36 =
(a - b + 6)(a + b - 6)
(a - b + 6)(a - b - 6)
(a + b - 6)(a + b + 6)
(a - b + 6)(a + b - 12)
Solution:
a
2
- b
2
+ 12b - 36 =
a
2
- (b
2
- 12b + 36)=
a
2
- (b - 6)
2
=
[a - (b - 6)][a + (b - 6)]=
(a - b + 6)(a + b - 6)
Easy
Normal
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