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Factoring Polynomials

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Factoring Polynomials: Difficult Problems with Solutions

By Catalin David

Problem 1

Factor 2x² + 7x + 5=

(2x + 1)(x - 5)

(x - 1)(2x + 5)

(x + 1)(2x + 5)

(x - 1)(2x - 5)

Solution:
2x² + 7x + 5 =
2x² + 2x + 5x + 5 =
2x ⋅ x + 2x ⋅ 1 + 5 ⋅ x + 5 ⋅ 1 =
2x(x + 1) + 5(x + 1) = (x + 1)(2x + 5)

Problem 2

Factor 3x² - 4x - 7 =

(3x - 7)(x - 1)

(-3x + 7)(x + 1)

(3x - 7)(x + 1)

(3x + 7)(x - 1)

Solution:
3x² - 4x - 7 =
3x² - 7x + 3x - 7 =
x ⋅ 3x + x ⋅ (-7) + 1 ⋅ 3x + 1 ⋅ (-7) =
x(3x - 7) + 1(3x - 7) = (3x - 7)(x + 1)

Problem 3

Factor -5x² + x + 6 =

(x + 2)(5x + 4)

-(x + 1)(-5x - 6)

(x + 1)(5x - 6)

(x + 1)(-5x + 6)

Solution:
-5x² + x + 6 =
-5x² -5x + 6x + 6 =
-5x ⋅ x - 5x ⋅ 1 + 6 ⋅ x + 6 ⋅ 1 =
-5x(x + 1) + 6(x + 1) = (x + 1)(-5x + 6)

Problem 4

Factor x⁴ + 2x² - 9x² - 18 =

(x² + 2)(x² - 9)

(x + 2)(x - 2)(x + 3)

(x² + 2)(x - 3)(x + 3)

(x² + 2)(x - 3)(x + 4)

Solution:
x⁴ + 2x² - 9x² - 18 =
x²⋅x² + x²⋅2 - 9⋅x²- 9⋅2 =
x²(x² + 2) - 9(x² + 2) =
(x² + 2)(x² - 9) =
(x² + 2)(x - 3)(x + 3)

Problem 5

Factor x⁴ + 3x² - 4x² - 12 =

(x² + 3)(x² - 4)

(x² + 3)(x + 2)(x - 1)

(x² + 3)(x - 1)(x - 2)

(x² + 3)(x + 2)(x - 2)

Solution:
x⁴ + 3x² - 4x² - 12 =
x²⋅x² + x²⋅3 - 4⋅x² - 4⋅3 =
x²(x² + 3) - 4(x² + 3) =
(x² + 3)(x² - 4) =
(x² + 3)(x - 2)(x + 2)

Problem 6

Factor (3x + 2)² + 9x + 6 =

(3x + 2)(3x + 1)

(3x + 2)(3x + 6)

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

(3x + 2)(3x + 5)

Solution:
(3x + 2)² + 9x + 6 =
(3x + 2)² + 3(3x + 2) =
(3x + 2)[(3x + 2) + 3] =
(3x + 2)(3x + 2 + 3) =
(3x + 2)(3x + 5)

Problem 7

Factor (2x-5)² + 4x - 10 =

(2x - 5)(2x - 5)

(2x - 5)(2x - 4)

(2x - 5)(2x - 3)

(2x - 5)(2x - 1)

Solution:
(2x - 5)² + 4x - 10 =
(2x - 5)² + 2(2x - 5) =
(2x - 5)[(2x - 5) + 2] =
(2x - 5)(2x - 5 + 2) =
(2x - 5)(2x - 3)

Problem 8

Factor (5x - 7)² - 15x + 21 =

-2(5x - 7)(x - 2)

5(5x - 7)(x - 2)

(5x - 7)(2x - 10)

(5x - 7)(5x - 5)

Solution:
(5x - 7)² - 15x + 21 =
(5x - 7)² - 3(5x - 7)=
(5x - 7)[(5x - 7)- 3]=
(5x - 7)(5x - 7 - 3)=
(5x - 7)(5x - 10) = (5x - 7)5(x - 2) = 5(5x - 7)(x - 2)

Problem 9

Factor (4x+1)² - 8x - 2 =

(4x + 1)(2x - 1)

(4x + 1)(-4x + 1)

(4x + 1)(4x - 1)

(4x + 3)(2x - 1)

Solution:
(4x + 1)² - 8x - 2 =
(4x + 1)² - 2(4x + 1) =
(4x + 1)[(4x + 1) - 2] =
(4x + 1)(4x + 1 - 2) =
(4x + 1)(4x - 1)

Problem 10

4x - 6 - (2x - 3)² =

(2x - 3)(5 - 2x)

(2x - 3)(5 - x)

(2x - 1)(4 - 2x)

(2x - 3)(6 - x)

Solution:
4x - 6 - (2x - 3)² =
2(2x - 3) - (2x - 3)² =
(2x - 3)[2 - (2x - 3)] =
(2x - 3)(2 - 2x + 3) =
(2x - 3)(5 - 2x)

Problem 11

Factor x² + 3x + 2=

(x + 1)(x + 4)

(x + 1)(x - 1)

(x + 1)(x + 3)

(x + 1)(x + 2)

Solution:
We can write: ax² + bx + c = ax² + dx + ex + c,
d⋅e = a⋅c and d + e = b

x² + 3x + 2 =
x² + x + 2x + 2 =
x⋅x + x⋅1 + 2⋅x + 2⋅1 =
x(x + 1) + 2(x + 1) =
(x + 1)(x + 2)

Problem 12

Factor x² + 6x + 8 =

(x + 2)(x + 8)

(x + 3)(x + 4)

(x + 2)(x + 4)

(x + 2)(x + 5)

Solution:
We can write: ax² + bx + c = ax² + dx + ex + c,
d⋅e = a⋅c and d + e = b

x² + 6x + 8 =
x² + 2x + 4x + 8 =
x⋅x + x⋅2 + 4⋅x + 4⋅1=
x(x + 2) + 4(x + 2) =
(x + 2)(x + 4)

Problem 13

Factor 9x - 12 + (3x - 4)²=

(5x - 4)(2x + 1)

(x - 4)(3x - 1)

(3x - 4)(3x + 1)

(3x - 4)(3x - 1)

Solution:
9x - 12 + (3x - 4)² =
3(3x - 4) + (3x - 4)² =
(3x - 4)[3 + (3x - 4)]=
(3x - 4)(3 + 3x - 4)=
(3x - 4)(3x - 1)

Problem 14

Factor x² + 6x + 8 =

(x + 4)(x + 2)

(x + 3)(x + 2)

(x + 4)(x - 2)

(x + 5)(x + 2)

Solution:
We can write: ax² + bx + c = ax² + dx + ex + c,
d⋅e = a⋅c and d + e = b

x² + 6x + 8 =
x² + 4x + 2x + 8 =
x⋅x + x⋅4 + 2⋅x + 2⋅4 =
x(x + 4) + 2(x + 4) = (x + 4)(x + 2)

Problem 15

Factor x² + 9x + 20 =

(x + 5)(x - 4)

(x + 5)(x + 4)

(x + 5)(x + 3)

(x + 2)(x + 4)

Solution:
We can write: ax² + bx + c = ax² + dx + ex + c,
d⋅e = a⋅c and d + e = b

x² + 9x + 20 =
x² + 5x + 4x + 20 =
x⋅x + x⋅5 + 4⋅x + 4⋅5=
x(x + 5) + 4(x + 5) = (x + 5)(x + 4)

Problem 16

Factor x² - 5x + 6 =

(x + 2)(x + 3)

(x - 2)(x + 3)

(x - 2)(x - 3)

(x - 2)(x - 5)

Solution:
We can write: ax² + bx + c = ax² + dx + ex + c,
d⋅e = a⋅c and d + e = b

x² - 5x + 6 =
x² - 3x - 2x + 6 =
x⋅x + x⋅(-3) - 2⋅x - 2⋅(-3) =
x(x - 3) - 2(x - 3) = (x - 2)(x - 3)

Problem 17

Factor x² - 8x + 15 =

(x - 3)(x - 4)

(x - 3)(x + 5)

(x - 4)(x + 2)

(x - 3)(x - 5)

Solution:
We can write: ax² + bx + c = ax² + dx + ex + c,
d⋅e = a⋅c and d + e = b

x² - 8x + 15 =
x² - 3x - 5x + 15 =
x⋅x + x⋅(-3) - 5⋅x -5⋅(-3) =
x(x - 3) - 5(x - 3) = (x - 3)(x - 5)

Problem 18

Factor x² - 7x + 12 =

(x - 4)(x - 3)

(x - 4)(x + 3)

(x + 4)(x + 3)

(x - 5)(x - 3)

Solution:
We can write: ax² + bx + c = ax² + dx + ex + c,
d⋅e = a⋅c and d + e = b

x² - 7x + 12 =
x² -4x - 3x + 12 =
x⋅x + x⋅(-4) - 3⋅x - 3⋅(-4) =
x(x - 4) - 3(x - 4) =
(x - 4)(x - 3)

Problem 19

Factor x² + x - 20 =

(x + 4)(x - 5)

(x - 4)(x + 5)

(x - 4)(x - 5)

(x + 4)(x + 5)

Solution:
We can write: ax² + bx + c = ax² + dx + ex + c,
d ⋅ e = a ⋅ c and d + e = b

x² + x - 20 =
x² - 4x + 5x - 20 =
x ⋅ x + x ⋅ (-4) + 5 ⋅ x + 5 ⋅ (-4) =
x(x - 4) + 5(x - 4) = (x - 4)(x + 5)

Problem 20

Factor x² - 4x - 12 =

(x - 2)(x + 4)

(x + 6)(x - 2)

(x - 6)(x + 2)

(x - 6)(x - 2)

Solution:
We can write: ax² + bx + c = ax² + dx + ex + c,
d ⋅ e = a ⋅ c and d + e = b

x² - 4x - 12 =
x² - 6x + 2x - 12 =
x ⋅ x + x ⋅(-6) + 2 ⋅ x + 2 ⋅(-12) =
x(x - 6) + 2(x - 6) = (x - 6)(x + 2)

Problem 21

Factor x² - x - 6 =

(x - 2)(x + 3)

(x - 2)(x + 4)

(x - 3)(x - 2)

(x - 3)(x + 2)

Solution:
We can write: ax² + bx + c = ax² + dx + ex + c,
d ⋅ e = a ⋅ c and d + e = b

x² - x - 6 =
x² - 3x + 2x - 6 =
x ⋅ x + x ⋅ (-3) + 2 ⋅ x + 2 ⋅ (-3) =
x(x - 3) + 2(x - 3) = (x - 3)(x + 2)

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