FACTORING SPECIAL PRODUCTS WORKSHEET

Problems 1-4 : Determine whether each trinomial is a perfect square. If so, factor. If not, explain.

Problem 1 : 

x2 + 12x + 36

Problem 2 : 

9x2 + 12x + 4

Problem 3 : 

4x2 - 12x + 9

Problem 4 : 

x2 + 9x + 16

Problems 5-8 : Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.

Problem 5 :

x2 - 81

Problem 6 : 

4x2 - 9

Problem 7 : 

9p4 - 16q2

Problem 8 : 

x6 - 7y2

Problems 9-12 : Factor the given trinomial.

Problem 9 : 

x2 + 6x + 9

Problem 10 : 

4y2 + 20y + 25

Problem 11 : 

81z2 - 18z + 1

Problem 12 : 

100k2 - 140k + 49

Problems 13-16 : Factor the given binomial.

Problem 13 :

x2 - 144

Problem 14 :

25y2 - 169

Problem 15 :

9a2 - 16b2

Problem 16 :

c4 - d4

Answers

1. Answer :

x2 + 12x + 36

The trinomial is a perfect square. Factor.

x2 + 12x + 36

a = x, b = 6.

Write the trinomial as a2 + 2ab + b2.

x2 + 2(x)(6) + 62

Write the trinomial as (a + b)2.

= (x + 6)2

2. Answer :

9x2 + 12x + 4

The trinomial is a perfect square. Factor.

9x2 + 12x + 4

a = 3x, b = 2. 

Write the trinomial as a2 + 2ab + b2.

= (3x)2 + 2(3x)(2) + 22

Write the trinomial as (a + b)2.

= (3x + 2)2

3. Answer :

4x2 - 12x + 9

The trinomial is a perfect square. Factor.

= 4x2 - 12x + 9

a = 2x, b = 3. 

Write the trinomial as a2 - 2ab + b2.

= (2x)2 - 2(2x)(3) + 32

Write the trinomial as (a - b)2.

= (2x - 3)2

4. Answer :

x2 + 9x + 16

2(x · 4)  ≠  9x

x2 + 9x + 16 is not a perfect-square trinomial because

9x ≠ 2(x · 4)

5. Answer :

x2 - 81

The polynomial is a difference of two squares.

= x2 - 81

x2 - 92

a = x and b = 9, write the polynomial as (a + b)(a - b).

=  (x + 9)(x - 9)

6. Answer :

4x2 - 9

The polynomial is a difference of two squares.

= 4x2 - 9

= (2x)2 - 32

a = 2x and b = 3, write the polynomial as (a + b)(a - b).

=  (2x + 3)(2x - 3)

7. Answer :

9p4 - 16q2

The polynomial is a difference of two squares.

= 9p4 - 16q2

= (3p2)2 - (4q)2

a = 3p2 and b = 4q, write the polynomial as (a + b)(a - b).

= (3p2 + 4q)(3p2 - 4q)

8. Answer :

x6 - 7y2

7y2 is not a perfect square.

x6 - 7yis not the difference of two squares because 7yis not a perfect square.

9. Answer :

= x2 + 6x + 9

= x2 + 2(x)(3) + 32

Since the above expression is in the form of a2 + 2ab + b2, it can be written in the form (a + b)2.

= (x + 3)2

10. Answer :

= 4y2 + 20y + 25

= 22y2 + 20y + 52

= (2y)2 + 2(2y)(5) + 52

Since the above expression is in the form of a2 + 2ab + b2, it can be written in the form (a + b)2.

= (2y + 5)2

11. Answer :

81z2 - 18z + 1

= 92z2 - 18z + 12

= 92z2 - 2(9z)(1) + 12

= (9z)2 - 2(9z)(1) + 12

Since the above expression is in the form of a2 - 2ab + b2, it can be written in the form (a - b)2.

= (9z - 1)2

12. Answer :

100k2 - 140k + 49

= 102k2 - 140k + 72

= (10k)2 - 2(10k)(7) + 72

Since the above expression is in the form of a2 - 2ab + b2, it can be written in the form (a - b)2.

= (10k - 7)2

13. Answer :

= x2 - 144

= x2 - 122

The above binomial is a difference of two squares and it is in the form of (a2 - b2). Take a = x and b = 12 and write the above binomial in the factored form (a + b)(a - b).

= (x + 12)(x - 12)

14. Answer :

= 25y2 - 169

= 52y2 - 132

= (5y)2 - 132

The above binomial is a difference of two squares and it is in the form of (a2 - b2). Take a = 5y and b = 13 and write the above binomial in the factored form (a + b)(a - b).

= (5y + 13)(5y - 13)

15. Answer :

= 9a2 - 16b2

= 32a2 - 42b2

= (3a)2 - (4b)2

The above binomial is a difference of two squares and it is in the form of (a2 - b2). Take a = 3a and b = 4b and write the above binomial in the factored form (a + b)(a - b).

= (3a + 4b)(3a - 4b)

16. Answer :

= c4 - d4

= (c2)2 - (d2)2

= (c2)2 - (d2)2

The above binomial is a difference of two squares and it is in the form of (a2 - b2). Take a = c2 and b = d2 and write the above binomial in the factored form (a + b)(a - b).

= (c2 + d2)(c2 - d2)

Since (c2 - d2) is a difference of two squares, it can be factored as (c + d)(c - d).

= (c2 + d2)(c + d)(c - d)

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