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# Special Products of Polynomials Worksheet

Special Products of Polynomials Worksheet
• Page 1
1.
Katie decided to expand her tulip flower garden which is in the shape of a square by increasing the length of its side by 5 feet. If $x$ represents the length of the present garden, which of the following expressions represents the difference between the area of her new garden and the area of the present garden?
 a. $x$2 + 10$x$ + 25 b. 4$x$ + 20 c. 10$x$ + 25 d. 2$x$ + 5

#### Solution:

Area of the original tulip flowers garden = x2
[Area of the square = (side)2.]

Area of the new tulip flowers garden = (x + 5)2
[Length is 5 feet longer than the original.]

Difference between the new area and the original area of the garden = (x + 5)2 - x2

= x2 + 10x + 25 - x2
[Use (a + b)2 = a2 + 2ab + b2.]

= 10x + 25
[Simplify.]

Correct answer : (3)
2.
Find the product:
(1 - $\frac{1}{{2}^{2}}$) (1 - $\frac{1}{{3}^{2}}$) (1 - $\frac{1}{{4}^{2}}$) . . . (1 - $\frac{1}{{2004}^{2}}$)
 a. $\frac{2004}{2005}$ b. 1 c. $\frac{2005}{4008}$

#### Solution:

(1 - 122) (1 - 132) (1 - 142) . . . (1 - 120042)

= (1 - 12) (1 + 12) (1 - 13) (1 + 13) . . . (1 - 12004) (1 + 12004)
[Use formula: a2 - b2 = (a - b) (a + b).]

= (1 - 12) (1 - 13) (1 - 14) . . . (1 - 12004) (1 + 12) (1 + 13) (1 + 14) . . . (1 + 12004)

= (1 / 2) (2 / 3) (3 / 4) . . . (2003 / 2004) (3 / 2 ) (4 / 3) (5 / 4) . . . (2005 / 2004)
[Simplify.]

= 12004×20052=20054008

Correct answer : (3)
3.
Which of the following expressions is the difference of two squares?
 a. $z$2 - 36 b. $z$2 - 11 c. $z$2 - 7 d. $z$2 - 27

#### Solution:

The expressions z2 - 7, z2 - 27, and z2 - 11 cannot be written as the difference of two squares, since 7, 27 and 11 are not perfect squares.

Consider the expression z2 - 36

= (z)2 - (6)2

So, the expression z2 - 36 is the difference of two squares.

Correct answer : (1)
4.
The edge of a cube is ($x$ + 5). Which expression represents its surface area?
 a. 6$x$2 + 60 b. ($x$ + 5)3 c. $x$2 + 10$x$ + 25 d. 6$x$2 + 60$x$ + 150

#### Solution:

Edge of the cube = (x + 5)

Surface area = 6 × (edge)2

= 6 × (x + 5)2

= 6(x2 + 10x + 25)
[Use (a + b)2 = a2 + 2ab + b2]

= 6x2 + 60x + 150

Correct answer : (4)
5.
Which of the following expressions is not the difference of two squares?
 a. $z$6 - 36 b. $z$2 + 64 c. $z$2 - 4 d. $z$2 - 9

#### Solution:

z2 - 4 = z2 - (2)2

z2 - 9 = z2 - (3)2

z6 - 36 = (z3)2 - (6)2

z2 + 64 = (z)2 + (8)2

So, z2 + 64 is not the difference of two squares.

Correct answer : (2)
6.
Which of the the following expressions is the square of a binomial?
 a. $a$2 - 24$a$ - 144 b. $a$2 + 18$a$ + 81 c. $a$2 + 18$a$ - 81 d. $a$2 - 18$a$ - 81

#### Solution:

a2 - 18a - 81 = (a)2 - 2(9)(a) - (9)2

≠ (a - 9)2
[(a - b)2 = a2 - 2ab + b2.]

So, the expression a2 - 18a - 81 is not the square of a binomial.

a2 + 18a - 81 = (a)2 + 2(9)(a) - (9)2

≠ (a + 9)2
[(a + b)2 = a2 + 2ab + b2.]

So, the expression a2 + 18a - 81 is not the square of a binomial.

a2 - 24a - 144 = (a)2 - 2(12)(a) - (12)2

≠ (a - 12)2
[(a - b)2 = a2 - 2ab + b2.]

So, the expression a2 - 24a - 144 is not the square of a binomial.

a2 + 18a + 81 = (a)2 + 2(9)(a) + (9)2

= (a + 9)2
[(a + b)2 = a2 + 2ab + b2.]

So, the expression a2 + 18a + 81 is the square of a binomial.

Correct answer : (2)
7.
Which of the the following expressions is the square of a binomial?
 a. $c$2 - 20$c$ - 100 b. $c$2 - 18$c$ - 81 c. $c$2 - 18$c$ + 81 d. $c$2 + 18$c$ - 81

#### Solution:

c2 - 18c - 81 = (c)2 - 2(9)(c) - (9)2

≠ (c - 9)2
[(a - b)2 = a2 - 2ab + b2.]

So, the expression c2 - 18c - 81 is not the square of a binomial.

c2 + 18c - 81 = (c)2 + 2(9)(c) - (9)2

≠ (c + 9)2
[(a + b)2 = a2 + 2ab + b2.]

So, the expression c2 + 18c - 81 is not the square of a binomial.

c2 - 20c - 100 = (c)2 - 2(10)(c) - (10)2

≠ (c - 10)2
[(a - b)2 = a2 - 2ab + b2.]

So, the expression c2 - 20c - 100 is not the square of a binomial.

c2 - 18c + 81 = (c)2 - 2(9)(c) + (9)2

= (c - 9)2
[(a - b)2 = a2 - 2ab + b2.]

So, the expression c2 - 18c + 81 is the square of a binomial.

Correct answer : (3)
8.
Which of the following expressions is not the square of binomial?
 a. $x$2 + 16$x$ + 64 b. $x$2 - 16$x$ + 64 c. $x$2 - 16$x$ - 64 d. $x$2 - 18$x$ + 81

#### Solution:

x2 - 16x + 64 = (x)2 - 2(x)(8) + (8)2

= (x - 8)2, which is square of a binomial.
[(a - b)2 = a2 - 2ab + b2.]

x2 + 16x + 64 = (x)2 + 2(x)(8) + (8)2

= (x + 8)2, which is square of a binomial.
[(a + b)2 = a2 + 2ab + b2.]

x2 - 18x + 81 = (x)2 - 2(x)(9) + (9)2

= (x - 9)2, which is square of a binomial.
[(a - b)2 = a2 - 2ab + b2.]

x2 - 16x - 64 = (x)2 - 2(x)(8) - (8)2

≠ (x - 8)2
[(a - b)2 = a2 - 2ab + b2.]

So, the expression x2 - 16x - 64 is not the square of a binomial.

Correct answer : (3)
9.
Which of the following expressions is the square of a binomial?
 a. $x$2 - 4$x$ - 4 b. $x$2 - 4$x$ - 16 c. $x$2 - 4$x$ + 4 d. $x$2 - 4$x$ + 16

#### Solution:

The expressions x2 - 4x + 16, x2 - 4x - 4 and x2 - 4x - 16 cannot be expressed as the square of a binomial.

x2 - 4x + 4 = (x - 2)2
[Use a2 - 2ab + b2 = (a - b)2]

So, the expression x2 - 4x + 4 is the square of a binomial.

Correct answer : (3)
10.
Find the product of ($c$ + 9) and ($c$ - 9).
 a. $c$2 - 81 b. $c$2 + 90 c. $c$2 - 9$c$ + 81 d. $c$2 - 9

#### Solution:

(a + b)(a - b) = a2 - b2
[Write pattern.]

(c + 9)(c - 9)

= (c)2 - (9)2
[Apply pattern.]

= c2 - 81
[Simplify.]

Correct answer : (1)

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