# Simplifying Rational Expressions – Explanation & Examples

Now that you understand what rational numbers are, the next topic to look at in this article is **rational expressions and how to simplify them**. Just for your own benefit, we define a rational number as a number expressed in the form of p/q where it is not equal to zero.

In other words, we can say a rational number is nothing more than a fraction in which the numerator and the denominator are integers. Examples of rational numbers are 5/7, 4/9/ 1/ 2, 0/3, 0/6 etc.

On the other hand, a rational expression is an algebraic expression of the form f(x) / g(x) in which the numerator or denominator are polynomials, or both the numerator and the numerator are polynomials.

Examples of rational expression are 5/x − 2, 4/(x + 1), (x + 5)/5, (x^{2 }+ 5x + 4)/(x + 5), (x + 1)/(x + 2), (x^{2 }+ x + 1)/2x etc.

## How to Simplify Rational Expressions?

Simplification of rational expression is the process of reducing a rational expression in its lowest terms possible. Rational expressions are simplified in the same way in which numerical numbers or fractions are simplified.

*To simplify any rational expressions, we apply the following steps:*

- Factorize both the denominator and numerator of the rational expression. Remember to write each expression in standard form.
- Reduce the expression by cancelling out common factors in the numerator and denominator
- Rewrite the remaining factors in the numerator and denominator.

*Let’s simplify a couple of examples as shown below:*

*Example 1*

Simplify: (x^{2 }+ 5x + 4) (x + 5)/(x^{2 }– 1)

__Solution__

Factoring the numerator and denominator to get;

⟹ (x + 1) (x + 4) (x + 5)/(x + 1) (x – 1)

Now cancel the common terms.

⟹ (x + 4) (x + 5)/(x – 1)

*Example 2*

Simplify (x^{2} – 4) / (x^{2}+ 4x + 4)

__Solution__

Factor both the numerator and denominator to get.

⟹ (x + 2) (x – 2) / (x + 2) (x + 2)

Now cancel out common factors in the numerator and denominator to get.

= (x – 2) / (x + 2)

*Example 3*

Simplify the rational expression x / (x^{2} – 4x)

__Solution__

Factor x out in the denominator to get;

⟹x /x (x – 4)

On cancelling the common terms in the top and bottom, we get;

= 1 / (x – 4)

*Example 4*

Simplify the rational expression (5x + 20) / (7x + 28)

__Solution__

Factor out the GCF in both the numerator and denominator;

= (5x + 20) / (7x + 28) ⟹ 5(x + 4) / 7(x + 4)

On cancelling common terms, we get;

= 5/7

*Example 5*

Simplify the rational expression (x^{2} + 7x + 10) / (x^{2} – 4)

__Solution__

Factor both the top and bottom of the expression.

= (x^{2} + 7x + 10) / (x^{2} – 4) ⟹ (x + 5) (x + 2) / (x^{2 }– 2^{2})

⟹ (x + 5) (x + 2) / (x + 2) (x – 2)

Cancel the common terms to get;

= (x + 5) / (x – 2)

*Example 6*

Simplify (3x + 9) / (3x + 15)

__Solution__

= (3x + 9) / (3x + 15) ⟹ 3(x + 3) / 3(x + 5)

= (x + 3) / (x + 5)

*Example 7*

Simplify the rational expression (64a^{3} + 125b^{3}) / (4a^{2}b + 5ab^{2})

__Solution__

Factor the numerator and the top;

= (64a^{3} + 125b^{3}) / (4a^{2}b + 5ab^{2}) ⟹ [(4a)^{3} + (5b)^{3}] / ab (4a + 5b)

⟹ (4a + 5b) [(4a)^{2} – (4a) (5b) + (5b)^{2}] / ab (4a + 5b)

Cancel out common terms to get;

= (16a^{2} – 20ab + 25b^{2}) / ab

*Example 8*

Simplify the following rational expression

(9x^{2} – 25y^{2}) / (3x^{2} – 5xy)

__Solution__

= (9x^{2} – 25y^{2}) / (3x^{2} – 5xy) ⟹ [(3x)^{2} – (5y)^{2}] / x (3x – 5y)

= [(3x + 5y) (3x – 5y)] / x (3x – 5y)

= (3x + 5y) / x

*Example 9*

Simplify: (6x^{2} – 54) / (x^{2} + 7x + 12)

__Solution__

= (6x^{2} – 54) / (x^{2} + 7x + 12)

= 6(x^{2} – 9) / (x + 3) (x + 4)

= 6(x^{2 }– 3^{2}) / (x + 3) (x + 4)

= 6(x + 3) (x – 3) / (x + 3) (x + 4)

= 6(x – 3) / (x + 4)

*Practice Questions*

Simplify the following rational expressions:

- 4x
^{3}/ 8x^{2} - (4x
^{3}+ 8x^{2})/2x - (7x
^{2}+ 28x)/ (x^{2}+ 8x + 16) - (4x
^{2}+ 4x + 1)/ (2x^{3}+ 11x^{2}+ 5x) - (x
^{2}+ 2x – 15)/ (x^{2}+ x – 12) - (x
^{3}+ 1)/ (x^{2}+ 7x + 6) - x
^{2 }+ 10x + 24/x^{3 }– x^{2 }− 20x - x + 3/x
^{2 }+ 12x + 27 - (x
^{3}+ 4x^{2}– 9x – 36)/ (4x^{2}+ 28x + 48) - (3x
^{2}– 9xy – 12y^{2})/ (6x^{3}– 6xy^{2}) - (2x
^{4}+ 9x^{3}-5x^{2})/ (6x^{3}+ x^{2}– 2x) - (2x
^{3 }+ 5x^{2 }+ 9)/ (2x^{2}– x + 3) - (x
^{3 }+ 3x^{2})/2x - (xy + 3x − 2y – 6)/ (y
^{2 +}y – 6) - (5m
^{2 }− 57mn + 70n^{2})/ 2m^{2}− 16mn − 40n^{2}

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