EXAMPLES OF SIMPLIFYING RATIONAL EXPRESSIONS

The expression which is in the form of f(x) / g(x) is called rational expression.

Simplifying rational expression is nothing but expressing the the rational expression to lowest term or simplest form.

The following steps ill be useful to simple rational expressions. 

Example :

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)  Answer :

(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)  Answer :

(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)  Answer :

(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)  Answer :

(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)  Answer :

(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)  Answer :

(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)   Answer :

(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)  Answer :

(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)  Answer :

[(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)  Answer :

[(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)  Answer :

(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).

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here. 

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.