Simplify algebraic fractions

In this chapter we will learn to simplify algebraic fraction to its lowest terms.

To understand the chapter you should have basic knowledge about the concept of fractions and polynomial factorization.

How to simplify algebraic fractions ?

The fractions which contain polynomial on numerator or denominator are called algebraic polynomials.

To simplify the algebraic fractions, follow the below steps;

(a) Factorize the numerator and denominator into smaller factors.

(b) Find the common factors between numerator and denominator and cancel them.

(c) After cancelling common factors, you will get algebraic factors in reduced form.

I hope you understood the above process. Let us now solve some problems for further clarity.

Simplify algebraic fraction to lowest terms – Solved example

Example 01
Simply the below fraction.

\mathtt{\Longrightarrow \ \frac{\left( x^{2} -16\right)}{\left( x^{2} +4x\right)}}

Given above is the algebraic fraction.

For simplification, factorize the numerator and denominator separately.

Factorizing the numerator.

\mathtt{\Longrightarrow \ \left( x^{2} -16\right)}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2} -4^{2}\right)}

Referring the formula;
\mathtt{\left( a^{2} -b^{2}\right) =( a-b)( a+b)}

Using the formula, we get;

\mathtt{\Longrightarrow \ \left( x^{2} -4^{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( x-4)( x+4)}

Now factorizing the denominator;

\mathtt{\Longrightarrow \ \left( x^{2} +4x\right)}\\\ \\ \mathtt{\Longrightarrow \ x( x+4)}

Writing the fraction in factorized form and cancel the common factors.

\mathtt{\Longrightarrow \ \frac{( x-4)( x+4)}{x.( x+4)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{( x-4)\cancel{( x+4)}}{x.\cancel{( x+4)}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{x-4}{x}}

Hence, \mathtt{\frac{x-4}{x}} is the factorized form of given fraction.

Example 02
Simplify the algebraic fraction.
\mathtt{\Longrightarrow \ \frac{\left( x^{2} +6x+9\right)( x+5)}{x^{3} +3x^{2}}}

To simplify the algebraic expression, follow below steps;

Factorize the numerator.

\mathtt{\Longrightarrow \left( x^{2} +6x+9\right)( x+5) \ }\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2} +2.x.3+3^{2}\right)( x+5)}

Referring the formula;

\mathtt{( a+b)^{2} =a^{2} +b^{2} +2ab}

Using the formula, we get;

\mathtt{\Longrightarrow \ ( x+3)^{2}( x+5)}

Now factorizing the denominator.

\mathtt{\Longrightarrow \ x^{3} +3x^{2}}\\\ \\ \mathtt{\Longrightarrow \ x^{2}( x+3)}

Writing the fraction in factorized form and cancelling the common factors

\mathtt{\Longrightarrow \ \frac{\mathtt{( x+3)^{2}( x+5)}}{\mathtt{\ x^{2}( x+3)}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\mathtt{( x+3)^{\cancel{2}} \ ( x+5)}}{\mathtt{\ x^{2}} \ \ \cancel{( x+3)}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{( x+3)( x+5)}{x^{2}}}

Hence, the given polynomial is factorized to \mathtt{\frac{( x+3)( x+5)}{x^{2}}} .

Example 03
Simplify the below fractions.
\mathtt{\Longrightarrow \ \frac{4+20x+3x^{2} +15x^{3}}{4x^{3} +20x^{4}}}

Factorizing the numerator

\mathtt{\Longrightarrow \ 4+20x+3x^{2} +15x^{3}}\\\ \\ \mathtt{\Longrightarrow \ 4\ ( 1+5x) +3x^{2}( 1+5x)}\\\ \\ \mathtt{\Longrightarrow \ ( 1+5x) \ \left( 4+3x^{2}\right)}

Factorizing the denominator

\mathtt{\Longrightarrow \ 4x^{3} +20x^{4}}\\\ \\ \mathtt{\Longrightarrow \ 4x^{3}( 1+5x)}

Writing the fraction in factorized form and cancelling the common terms.

\mathtt{\Longrightarrow \ \frac{\mathtt{( 1+5x) \ \left( 4+3x^{2}\right)}}{\mathtt{4x^{3}( 1+5x)}}}\\\ \\ \mathtt{\Longrightarrow \frac{\cancel{( 1+5x)}\mathtt{\ \left( 4+3x^{2}\right)}}{\mathtt{4x^{3}}\cancel{( 1+5x)}}}\\\ \\ \mathtt{\Longrightarrow \frac{\mathtt{\left( 4+3x^{2}\right)}}{\mathtt{4x^{3}}}}

Hence, the given fraction is simplified to \mathtt{\frac{\mathtt{\left( 4+3x^{2}\right)}}{\mathtt{4x^{3}}}}

Example 04
Simplify the algebraic fraction.

\mathtt{\Longrightarrow \ \frac{21x^{3} y^{2} z^{4}}{7x^{2} y+49x^{3} z}}

Factorize the numerator.

\mathtt{\Longrightarrow \ 21x^{3} y^{2} z^{4}}

The numerator is already present in factorized form. No need to do anything here.

Factorize the denominator

\mathtt{\Longrightarrow \ 7x^{2} y+49x^{3} z}\\\ \\ \mathtt{\Longrightarrow 7x^{2}( y+7xz)}

Writing fraction in factorized form and cancelling out the common factors.

\mathtt{\Longrightarrow \ \frac{\mathtt{21x^{3} y^{2} z^{4}}}{\mathtt{7x^{2}( y+7xz)}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3\ \cancel{21} .\ \mathtt{x^{3-2} .\ y^{2} .z^{4}}}{\cancel{7\ } \ \cancel{x^{2}}\mathtt{( y+7xz)}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3xy^{2} z^{4}}{y+7xz}}

Hence, the fraction is simplified to \mathtt{\Longrightarrow \ \frac{3xy^{2} z^{4}}{y+7xz}}

Example 05
Simplify the algebraic fraction

\mathtt{\Longrightarrow \ \frac{\left( x^{2} -14x+49\right)( y+2)^{2}}{( x-7)^{4}\left( y^{2} -4\right)}}

Factorizing the numerator.

\mathtt{\Longrightarrow \ \left( x^{2} -14x+49\right)( y+2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2} -2.7.x+7^{2}\right)( y+2)^{2}}

Referring the formula;

\mathtt{( a-b)^{2} =a^{2} -2ab+b^{2}}

Using the formula, we get;

\mathtt{\Longrightarrow \ ( x-7)^{2} .\ ( y+2)^{2}}

Now factorizing the denominator

\mathtt{\Longrightarrow \ ( x-7)^{4}\left( y^{2} -4\right)}

Referring to formula;
\mathtt{a^{2} -b^{2} =\ ( a-b)( a+b)}

Using the formula, we get;

\mathtt{\Longrightarrow \ ( x-7)^{4\ }( y-2)( y+2)}

Writing the factorized fraction, we get;

\mathtt{\Longrightarrow \ \frac{( x-7)^{2}( y+2)^{2}}{\mathtt{( x-7)^{4\ }( y-2)( y+2)}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\cancel{( x-7)^{2}} .( y+2)^{2-1}}{\mathtt{( x-7)^{4-2\ } .( y-2) .}\cancel{( y+2)}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{( y+2)}{( x-7)^{2} .( y-2)}}

Hence, the polynomial is factorized to \mathtt{\ \frac{( y+2)}{( x-7)^{2} .( y-2)}}

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