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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.
x2 + 10x + 25
b.
4x + 20
c.
10x + 25
d.
2x + 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 - 122) (1 - 132) (1 - 142) . . . (1 - 120042)
a.
20042005
b.
1
c.
20054008


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.
z2 - 36
b.
z2 - 11
c.
z2 - 7
d.
z2 - 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.
6x2 + 60
b.
(x + 5)3
c.
x2 + 10x + 25
d.
6x2 + 60x + 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.
z6 - 36
b.
z2 + 64
c.
z2 - 4
d.
z2 - 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.
a2 - 24a - 144
b.
a2 + 18a + 81
c.
a2 + 18a - 81
d.
a2 - 18a - 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.
c2 - 20c - 100
b.
c2 - 18c - 81
c.
c2 - 18c + 81
d.
c2 + 18c - 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.
x2 + 16x + 64
b.
x2 - 16x + 64
c.
x2 - 16x - 64
d.
x2 - 18x + 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.
x2 - 4x - 4
b.
x2 - 4x - 16
c.
x2 - 4x + 4
d.
x2 - 4x + 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.
c2 - 81
b.
c2 + 90
c.
c2 - 9c + 81
d.
c2 - 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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