SIMPLIFYING POLYNOMIALS WITH DIVISION PRACTICE WORKSHEET

Simplify the following :

(1)  [(x2-2x)/(x+2)]  [(3x+6)/(x-2)]

(2)  [(x2-81)/(x2-4)]  [(x2+6x+8)/(x2-5x-36)]

(3)  [(x2-3x-10)/(x2-x-20)]  [(x2-2x+4)/(x3+8)]

(4)  [(x2-16)/(x2-3x+2)]  [(x2-4)/(x3+64)]  

[(x2-4x+16)/(x2-2x-8)]

(5)  [(3x2+2x-1)/(x2-x-2)] [(2x2-3x-2)/(3x2+5x-2)]

(6)  [(2x-1)/(x2+2x+4)] [(x4-8x)/(2x2+5x-3)] 

[(x+3)/(x2-2x)]

(7)  [(a+b)/(a-b)] [(a3-b3)/(a3+b3)]

(8)  [(x2-9y2)/(3x-3y)]  [(x2-y2)/(x2+4xy+3y2)]

(9)  [(x2-4x-12)/(x2-3x-18)]  [(x2-2x-3)/(x2+3x+2)]

(10)  [(x2-3x-10)/(x2-x-20)][(x2-4x+16)/(x3+64)]

(11)   [(x2-16)/(x-2)] [(x2-4)/(x3+64)]

(12)  [(x + 7)/(x2+14x+49)] [(x2+8x+7)/(x+1)]

Detailed Answer Key

Problem 1 :

[(x2-2x)/(x+2)]  [(3x+6)/(x-2)]

Solution :

Let f(x)  =   [(x2-2x)/(x+2)]  [(3x+6)/(x-2)]

f(x)  =  [(x2-2x)/(x+2)]  [(3x+6)/(x-2)]

f(x)  =  [x(x-2)/(x+2)]  [3(x+2)/(x-2)]

f(x)  =  3x

So, the value of f(x) is 3x.

Problem 2 :

[(x2-81)/(x2-4)]  [(x2+6x+8)/(x2-5x-36)]

Solution :

Let f(x)  =  [(x2-81)/(x2-4)]  [(x2+6x+8)/(x2-5x-36)]

x- 81  =  x2- 92  ==>  (x+9)(x-9)

x- 4  =  x2- 22  ==> (x+2)(x-2)

x2+6x+8  =  (x+2)(x+4)

x2-5x-36  =  (x-9)(x+4)

f(x)  =  [(x+9)(x-9)/(x+2)(x-2)] ⋅ [(x+2)(x+4)/(x-9)(x+4)]

By simplifying (x+9)/(x-2)

So, the value of f(x) is (x+9)/(x-2).

Problem 3 :

[(x2-3x-10)/(x2-x-20)]  [(x2-2x+4)/(x3+8)]

Solution :

Let f(x)  =  [(x2-3x-10)/(x2-x-20)]  [(x2-2x+4)/(x3+8)]

x2-3x-10  =  (x-5)(x+2)

x2-x-20  =  (x-5)(x+4)

a3+b =  (a+b)(a2-ab+b2)

x3+2 =  (x+2)(x2-2x+4)

By applying the factors in f(x), we get

 =  [(x-5)(x+2)/(x-5)(x+4) [(x2-2x+4)/(x+2)(x2-2x+4)]

=  1/(x+4)

So, the value of f(x) is 1/(x+4)

Problem 4 :

[(x2-16)/(x2-3x+2)]  [(x2-4)/(x3+64)]  

[(x2-4x+16)/(x2-2x-8)]

Solution :

Let f(x)  =  [(x2-16)/(x2-3x+2)]  [(x2-4)/(x3+64)]  

[(x2-4x+16)/(x2-2x-8)]e

x2-16  =  x2-4 ==>  (x+4)(x-4)

x2-3x+2  =  (x-1)(x-2)

x2-4  =  x2-22  ==>  (x+2)(x-2)

x3+64  =  x3+4 ==> (x+4)(x2-4x+16) 

x2-2x-8  =  (x-4)(x+2)

=  [(x+4)(x-4)/(x-1)(x-2)][(x+2)(x-2)/(x+4)(x2-4x+16)]

[(x2-4x+16)/(x-4)(x+2)]

f(x)  =  1/(x-1)

So, the value of f(x) is 1/(x-1).

Problem 5 :

[(3x2+2x-1)/(x2-x-2)] [(2x2-3x-2)/(3x2+5x-2)]

Solution :

Let f(x)  =  [(3x2+2x-1)/(x2-x-2)]

 [(2x2-3x-2)/(3x2+5x-2)]

(3x2+2x-1)  =  (3x-1) (x+1)

(x2-x-2)  =  (x-2) (x+1)

(2x2-3x-2)  =  (2x+1) (x-2)

(3x2+5x-2) =   (2x-1) (x+2)

By applying the factors in f(x), we get

=  [(3x-1)(x+1)/(x-2) (x+1)][(2x+1) (x-2)/(2x-1) (x+2)]

=  (2x+1)/(x+2)

So, the value of f(x) is (2x+1)/(x+2).

Problem 6 :

[(2x-1)/(x2+2x+4)] [(x4-8x)/(2x2+5x-3)] 

[(x+3)/(x2-2x)]

Solution :

Let f(x)  =  [(2x-1)/(x2+2x+4)] [(x4-8x)/(2x2+5x-3)] 

[(x+3)/(x2-2x)]

x4-8x  =  x(x3-23)

x4-8x  =  x(x-2)(x2+2x+4)

2x2+5x-3  =  (2x-1)(x+3)

x2-2x  =  x(x-2)

By applying the factors in f(x), we get

=  [(2x-1)/(x2+2x+4)][x(x-2)(x2+2x+4)/(2x-1)(x+3)] 

[(x+3)/x(x-2)]

=  1

So, the value of f(x) is 1.

Problem 7 :

[(a+b)/(a-b)] [(a3-b3)/(a3+b3)]

Solution :

Let f(x)  =  [(a+b)/(a-b)] [(a3-b3)/(a3+b3)]

=  [(a+b)/(a-b)][(a-b)(a2+ab+b2)/(a+b) (a2-ab+b2)]

=  (a2+ab+b2)/(a2-ab+b2)

So, the value of f(x) is (a2+ab+b2)/(a2-ab+b2).

Problem 8 :

[(x2-9y2)/(3x-3y)]  [(x2-y2)/(x2+4xy+3y2)]

Solution :

Let f(x)  =  [(x2-9y2)/(3x-3y)]  [(x2-y2)/(x2+4xy+3y2)]

x2-9y=  x2-(3y)2

x2-9y2  =  (x+3y)(x-3y)

3x-3y  =  3(x-y)

x2-y=  (x+y)(x-y)

x2+4xy+3y=  (x+3y)(x+y)

By applying the factors in f(x), we get

=  [(x+3y)(x-3y)/3(x-y)]⋅[(x+y)(x-y)/(x+3y)(x+y)]

By simplifying, we get

=  (x-3y)/3

So, the value of f(x) is (x-3y)/3.

Problem 9 :

[(x2-4x-12)/(x2-3x-18)]  [(x2-2x-3)/(x2+3x+2)]

Solution :

Let f(x)  =  [(x2-4x-12)/(x2-3x-18)] 

 [(x2-2x-3)/(x2+3x+2)]

x2-4x-12  =  (x-6)(x+2)

x2-3x-18  =  (x-6)(x+3)

x2-2x-3  =  (x-3)(x+1)

x2+3x+2  =  (x+1)(x+2)

f(x)  =  [(x-6)(x+2)/(x-6)(x+3)]⋅[(x-3)(x+1)/(x+1)(x+2)]

f(x)  =  (x-3)/(x+3)

So, the value of f(x) is (x-3)/(x+3).

Problem 10 :

[(x2-3x-10)/(x2-x-20)][(x2-4x+16)/(x3+64)]

Solution :

Let f(x)  =  [(x2-3x-10)/(x2-x-20)][(x2-4x+16)/(x3+64)]

x2-3x-10  =  (x-5)(x+2)

x2-x-20  =  (x-5)(x+4)

x3+43  =  (x+4)(x2-4x+16)

By applying the factors in f(x), we get

f(x)  =  [(x-5)(x+2)/(x-5)(x+4)][(x2-4x+16)/(x+4)(x2-4x+16)]

f(x)  =  (x+2)/(x+4)2

So, the value of f(x) is (x+2)/(x+4)2.

Problem 11 :

 [(x2-16)/(x-2)] [(x2-4)/(x3+64)]

Solution :

Let f(x)  =  [(x2-16)/(x-2)] [(x2-4)/(x3+64)]

x2-16  =  x2-42  ==>  (x+4)(x-4)

x2-4  =  x2-22  ==>  (x+2)(x-2)

x3+64  =  x3+43  ==>  (x+4)(x2-4x+16)

f(x)  =  [(x+4)(x-4)/(x-2)] [(x+2)(x-2)/(x+4)(x2-4x+16)]

f(x)  =  (x-4)(x-2)/(x2-4x+16)

So, the value of f(x) is (x-4)(x-2)/(x2-4x+16).

Problem 12 :

[(x + 7)/(x2+14x+49)] [(x2+8x+7)/(x+1)]

Solution :

Let f(x)  =  [(x + 7)/(x2+14x+49)] [(x2+8x+7)/(x+1)]

x2+14x+49  =  (x+7)(x+7)

x2+8x+7  =  (x+1)(x+7)

By applying the factors in f(x), we get

f(x)  =  [(x+7)/(x+7)(x+7)] [(x+1)(x+7)/(x+1)]

f(x)  =  1

So, the value of f(x) is 1.

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