CHOOSING A FACTORING METHOD WORKSHEET

Problem 1 :

x² + 2x + xy + 2y

Problem 2 :

2ax² + bx² - 2ay² - by²

Problem 3 :

x³ + x² - x - 1

Problem 4 :

2x² - 8

Problem 5 :

2x - 2xy²

Problem 6 :

3t³ - 27t

Problem 7 :

4x⁴ - 4x²

Problem 8 :

3x + x² - 10

Problem 9 :

x³ + 8x² - x - 8

Problem 10 :

4a³ - 49a

Problem 11 :

2y² - 16y + 32

Problem 12 :

p² q - 25q + 3q² - 75

Problem 13 :

16 - w⁴

Problem 14 :

64x³ + 27

Problem 15 :

3t3 - 27t

Detailed Answer Key

Problem 1 :

x² + 2x + xy + 2y

Solution :

= x² + 2x + xy + 2y

Since we have four terms in the given expression, we have to use the method of grouping and do the factoring.

Factoring x from first two terms and factoring y from third and fourth terms.

= x (x + 2) + y (x + 2)

= (x + y)(x + 2)

So, the factored form of the given expression is 

(x + y)(x + 2)

Problem 2 :

2ax² + bx² - 2ay² - by²

Solution :

= 2ax² + bx² - 2ay² - by²

Since we have four terms in the given expression, we have to use grouping method.

Factoring x² from first and second term, factoring y² from third and fourth term.

= x² (2a + b) - y² (2a + b)

= (x² - y²) (2a + b) 

So, the factored form of the given expressions is

(x² - y²) (2a + b)

Problem 3 :

x³ + x² - x - 1

Solution :

= x³ + x² - x - 1

Since we have four terms in the given expression, we have to use grouping method.

Factoring x² from first and second term, factoring -1 from third and fourth terms.

= x² (x + 1) - 1(x + 1)

= (x²  - 1)(x + 1)

Here (x²  - 1) looks like a² - b², by expanding this using algebraic identity, we get (a + b) (a - b)

= (x + 1)(x - 1)(x + 1)

Since x + 1 is repeating twice, we write in the exponential form

= (x + 1)²(x - 1)

Problem 4 :

2x² - 8

Solution :

= 2x² - 8

= 2(x² - 4)

= 2(x² - 2²)

= 2(x + 2)(x - 2)

So, the factored form of the expression is 2(x + 2)(x - 2).

Problem 5 :

2x - 2xy²

Solution :

= 2x - 2xy²

Factoring 2x, we get

= 2x (1 - y²)

Here 1 - y², looks like a² - b² then (a + b) (a - b)

= 2x(1 + y)(1 - y)

Problem 6 :

3t³ - 27t

Solution :

= 3t³ - 27t

Factoring 3t, we get

= 3t(t² - 9)

= 3t(t² - 3²)

Here t² - 3², looks like a² - b² then (a + b) (a - b)

= 3t (t + 3)(t - 3)

So, the factored form of the given expression is 

3t (t + 3)(t - 3)

Problem 7 :

4x⁴ - 4x²

Solution :

Factoring 4x2 from these two terms

= 4x² (x²  - 1)

Here x² - 1², looks like a² - b² then (x + 1)(x - 1)

= 4x² (x + 1)(x - 1)

Problem 8 :

3x + x² - 10

Solution :

3x + x² - 10

It is a trinomial, but it is not in the standard form.

= x² + 3x - 10

By splitting the middle term, we get

= x² + 5x - 2x - 10

= x(x + 5) - 2(x + 5)

= (x - 2)(x + 5)

So, the factored form of the given expression is 

(x - 2)(x + 5)

Problem 9 :

x³ + 8x² - x - 8

Solution :

= x³ + 8x² - x - 8

= x²(x + 8) - 1(x + 8)

= (x² - 1)(x + 8)

= (x + 1)(x - 1)(x + 8)

So, the factored form of the given expression is 

(x + 1)(x - 1)(x + 8)

Problem 10 :

4a³ - 49a

Solution :

= 4a³ - 49a

Factoring a from the given expression.

= a(4a² - 49)

= a((2a)² - 7²)

= a(2a + 7)(2a - 7)

So, the factored form of the given expression is

a(2a + 7)(2a - 7)

Problem 11 :

2y² - 16y + 32

Solution :

= 2y² - 16y + 32

Factoring 2, we get

= 2(y² - 8y + 16)

= 2(y² - 4y - 4y + 16)

= 2[y(y - 4) - 4(y - 4)]

= 2(y - 4)(y - 4)

So, the factored form of the given expression is 

 2(y - 4)(y - 4)

Problem 12 :

p² q - 25q + 3q² - 75

Solution :

= p² q - 25q + 3q² - 75

= q(p² - 25) + 3(p² - 25)

= (q + 3)(p² - 25)

Here (p² - 25) looks like a² - b², expanding this using algebraic identity, we get

= (q + 3)(q + 5)(q - 5)

Problem 13 :

16 - w⁴

Solution :

= 16 - w⁴

= 4² - (w²)²

= (4 + w²) (4 - w²)

= (4 + w²) (2² - w²)

= (4 + w²) (2 + w)(2 - w)

So, the factored form of the given expression is 

(4 + w²) (2 + w)(2 - w)

Problem 14 :

64x³ + 27

Solution :

= 64x³ + 27

= (4x)³ + 3³

Looks like a³ + b³ = (a + b) (a² - ab + b²)

= (4x + 3)((4x)² - 4x(3) + 3²)

= (4x + 3)(16x² - 12x + 9)

Problem 15 :

3t³ - 27t

Solution :

= 3t³ - 27t

= 3t(t² - 9)

= 3t(t² - 3²)

= 3t(t + 3)(t - 3)

So, the factored form of the expression is 

3t(t + 3)(t - 3)

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