SIMPLIFYING POLYNOMIAL EXPRESSIONS IN FRACTIONS

Example 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.

Example 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)]

x² - 81 = x²- 9²  ==> (x + 9)(x - 9)

x² - 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).

Example 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 + b³ = (a + b)(a² - ab + b²)

x³ + 2³ = (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)

Example 4 :

[(x² - 16)/(x^{2 -} 3x + 2)] ⋅ [(x² - 4)/(x³ + 64)] ⋅ 

[(x² - 4x + 16)/(x² - 2x - 8)]

Solution :

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

[(x² - 4x + 16)/(x² - 2x - 8)]e

x² - 16 = x² - 4²  ==> (x + 4)(x - 4)

x² - 3x + 2 = (x - 1)(x - 2)

x² - 4 = x² - 2²  ==> (x + 2)(x - 2)

x³ + 64  = x³ + 4³  ==> (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).

Example 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).

Example 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.

Example 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).

Example 8 :

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

Solution :

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

x²-9y²  =  x²-(3y)²

x²-9y²  =  (x+3y)(x-3y)

3x-3y  =  3(x-y)

x²-y²  =  (x+y)(x-y)

x²+4xy+3y²  =  (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.

Example 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).

Example 10 :

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

Solution :

Let f(x)  =  [(x ² - 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.

Example 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).

Example 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.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.