SIMPLIFYING RATIONAL EXPRESSIONS WORKSHEET

Simplify the following rational expression into their lowest forms.

(i)  (6x²+9x)/(3x²-12x)

(ii)  (x²+1)/(x⁴-1)

(iii)  (x³-1)/(x²+x+1)

(iv)  (x³-27)/(x²-9)

(v)  (x⁴+x²+1)/(x²+x+1)

(vi)  (x³+8)/(x⁴+4x²+16)

(vii)  (2x²+x-3)/(2x²+5x+3)

(viii)  (2x⁴-162)/(x²+9) (2x-6)

(ix)  [(x-3) (x²-5x+4)] / [(x-4) (x²-2x-3)]

(x)  [(x-8)(x²+5x-50)]/[(x+10)(x²-13x+40)]

(xi)  (4x²+9x+5)/(8x²+6x-5)

(i)  Solution :

(6x²+9x)/(3x²-12x)

Let f(x)  =  (6x²+9x)/(3x²-12x)

We may factor 3x from the numerator and denominator.

f(x)  =  3x(2x+3)/3x(x-4)

f(x)  =  (2x+3)/(x-4)

So, the value f(x) is (2x+3)/(x-4).

(ii)  Solution :

(x²+1)/(x⁴-1)

Let f(x)  =  (x²+1)/(x⁴-1)

f(x)  =  (x²+1)/((x²)²-(1²)²)

Using the algebraic identity a²-b², we may expand

a²-b²  =  (a+b)(a-b)

(x²)²-(1²)²  =  (x²+1)(x²-1)

(x²)²-(1²)²  =  (x²+1)(x+1)(x-1)

By applying factors in f(x), we get

f(x)  =  (x²+1)/(x²+1)(x+1)(x-1)

f(x)  =  1/(x+1)(x-1)

f(x)  =  1/(x²-1)

So, the value of f(x) is 1/(x²-1).

(iii)  Solution :

(x³-1)/(x²+x+1)

Let f(x)  =  (x³-1)/(x²+x+1)

Now we are going to use the following algebraic formula

a³-b³  =  (a-b) (a²+ab+b²)

x³-1³  =  (x-1)(x²+x+1)

By applying factors in f(x), we get

f(x)  =  (x-1)(x²+x+1)/(x²+x+1)

f(x)  =  (x-1)

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

(iv)  Solution :

(x³-27)/(x²-9)

Let f(x)  =  (x³-27)/(x²-9)

f(x)  =  (x³-3³)/(x²-3²)

Now we are going to use the following algebraic formula

a³-b³  =  (a-b) (a²+ab+b²)

a²-b²  =  (a+b) (a-b)

x³-3³  =  (x-3)(x²+3x+9)

x²-3²  =  (x+3) (x-3)

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

f(x)  =  [(x-3)(x²+3x+9)]/[(x+3) (x-3) ]

f(x)  =  (x²+3x+9)/(x+3)

So, the value of f(x) is (x²+3x+9)/(x+3).

(v)  Solution :

(x⁴+x²+1)/(x²+x+1)

Let f(x)  = (x⁴+x²+1)/(x²+x+1)

Now we are going to use the following algebraic formula

(x⁴+x²+1)  =  (x²+1)² - x²

f(x)  =  [(x²+1)²-x²]/(x²+x+1)

By applying factors in f(x), we get 

f(x)  =  [(x²+1+x) (x²+1-x)]/(x²+x+1)

f(x)  =  [(x²+x+1 ) (x²-x+1)]/(x²+x+1)

f(x)  =  (x²-x+1)

So, the value of f(x) is (x²-x+1).

(vi)  Solution :

(x³+8)/(x⁴+4x²+16)

Let f(x)  =  (x³+8)/(x⁴+4x²+16)

x³+8  =  x³+2³

x³+2³  =  (x+2)(x²+2x+4)

(x⁴+4x²+16)  =  (x²+4)²-(2x)²

(x⁴+4x²+16)  =  [(x²+4)+(2x)][(x²+4)-(2x)]

(x⁴+4x²+16)  =  (x²+2x+4)(x²-2x+4)

By applying the factors, we get

=  (x+2)(x²+2x+4)/(x²+2x+4)(x²-2x+4)

=  (x+2)/(x²-2x+4)

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

(vii)   Solution :

(2x²+x-3)/(2x²+5x+3)  

                       Let f(x)  = (2x²+x-3)/(2x²+5x+3)

2x²+x-3  =  (x-1) (2x+3)

2x²+5x+3  =  (x+1)(2x+3)

By applying the factored form in f(x), we get

f(x)  =  (x-1) (2x+3)/(x+1)(2x+3)

f(x)  =  (x-1)/(x+1)

So, the value of f(x) is (x-1)/(x+1).

(viii)  Solution :

(2x⁴-162)/(x²+9) (2x-6)

Let f(x)  =  (2x⁴-162)/(x²+9) (2x-6)

2x⁴-162  =  2(x⁴-81)

=  2((x²)²-(9²)²)

=  2(x²+9) (x²-9)

2x⁴-162  =  2(x²+9) (x²-3²)

2x⁴-162  =  2(x²+9) (x+3)(x-3)

2x-6  =  2(x-3)

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

f(x)  =  2(x²+9) (x+3)(x-3)/(x²+9) 2(x-3)

f(x)  =  x+3

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

(ix)  Solution :

[(x-3) (x²-5x+4)] / [(x-4) (x²-2x-3)]

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

x²-5x+4  =  (x-1)(x-4)

x²-2x-3  =  (x-3)(x+1)

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

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

By simplifying, we get

f(x)  =  (x-1)/(x+1)

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

(x)  Solution :

[(x-8)(x²+5x-50)]/[(x+10)(x²-13x+40)]

Let f(x)  =  [(x-8)(x²+5x-50)]/[(x+10)(x²-13x+40)]

x²+5x-50  =  (x+10)(x-5)

(x²-13x+40)  =  (x-8)(x-5)

f(x)  =  [(x-8)(x+10)(x-5)]/[(x+10)(x-8)(x-5)]

f(x)  =  1

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

(xi)  Solution :

(4x²+9x+5)/(8x²+6x-5)

Let f(x)  =  (4x²+9x+5)/(8x²+6x-5)

4x²+9x+5  =  (x+1) (4x+5)

8x²+6x-5  =  (4x+5) (2x-1)

f(x)  = (x+1)/(2x-1)

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

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