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Algebraic Fractions

Algebraic Fractions are the fractions using a variable in the numerator or denominator. For example $\frac{x}{2}$, the upper number in fraction, x (variable), is numerator and lower number, 2, is denominator. If fraction using a variable in the denominator have a certain restrictions. The denominator can never be equal to zero. Algebraic fractions have properties which are the same as those for numerical fractions, the only difference being that the the numerator and denominator are both algebraic expressions. We can add, subtract, multiply and divide fractions in algebra in the same way that we do in simple arithmetic.

Algebraic Fractions are fractions using variable in the numerator or in the denominator.

(1) Fraction using variable in the numerator only, are in the form,

$\frac{x}{3}$, $\frac{y + 1}{5}$

(2) Fractions using variable in the denominator are in the form,

$\frac{3}{x}$, x $\neq$ 0

$\frac{2 + x}{x - 3}$, x - 3 $\neq$ 0 or x $\neq$ 3


$\frac{5}{xy}$, x $\neq$ 0 and y $\neq$ 0

No values can be assigned to variables that create a denominator zero.

Where x and y are variables.

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Simplifying Algebraic Fractions

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Simplification of fraction means to make the fraction as simple as possible.
The main objective of simplifying algebraic fraction is, express rational numbers in reduced form. To simplifying algebraic fractions we can use the factoring techniques to factor numerators and denominators. Algebraic fractions are simplifying in the same way as the fractions.

Simplifying Algebraic Fractions

Solved Examples

Question 1: Simplify $\frac{x^2 + 4x}{x^2 - 16}$
Solution:
 
Given $\frac{x^2 + 4x}{x^2 - 16}$

Step 1: Factorize the expressions

x2 + 4x = x(x + 4)

and

x2 - 16 = x2 - 42 = (x - 4)(x + 4)

using identity a2 - b2 = (a - b)(a + b)

Step 2: Cancel the factors common to the numerator and denominator

$\frac{x^2 + 4x}{x^2 - 16}$ = $\frac{x(x + 4)}{(x - 4)(x + 4)}$

= $\frac{x}{x - 4}$
 

Question 2: Simplify $\frac{x^2 - 81}{x + 9}$
Solution:
 
Given $\frac{x^2 - 81}{x + 9}$

Step 1: Factorize the expression

x2 - 81 = x2 - 92 = (x - 9)(x + 9)

using identity a2 - b2 = (a - b)(a + b)

Step 2:
$\frac{x^2 - 81}{x + 9}$ = $\frac{(x - 9)(x + 9)}{x + 9}$

= x - 9 (by canceling).

 

Question 3: Simplify $\frac{16 - 8m}{m - 2}$
Solution:
 
Given $\frac{16 - 8m}{m - 2}$

Step 1: Factorize the expression

16 - 8m = 8(2 - m)

Step 2:
$\frac{16 - 8m}{m - 2}$ = $\frac{8(2 - m)}{m - 2}$

= $\frac{- 8(m - 2)}{m - 2}$

= - 8 (by canceling)
 

Multiplying Algebraic Fractions

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To multiply algebraic fractions, we multiply their numerators together
and then multiply their denominators together.

If $\frac{x}{y}$ and $\frac{m}{n}$ are two rational expressions with y $\neq$ 0 and n $\neq$ 0,

then $\frac{x}{y}$ * $\frac{m}{n}$ = $\frac{x \times m}{y \times n}$ = $\frac{xm}{yn}$.
Steps for Multiplying Algebraic Fractions:

Step 1: Factorize the numerators and denominators.

Step 2: Cancel the factors common to the numerator and denominator.

Step 3: Multiply the remaining numerators together and denominators together.

Solved Examples

Question 1: Solve $\frac{2m}{3n}$ * $\frac{5n}{4m}$
Solution:
 
Given $\frac{2m}{3n}$ * $\frac{5n}{4m}$

Step 1:
$\frac{2m}{3n}$ * $\frac{5n}{4m}$ = $\frac{2m\times5n}{3n\times4m}$

Step 2: Cancel the factors common to the numerator and denominator

$\frac{2m}{3n}$ * $\frac{5n}{4m}$ = $\frac{2\times5\times m\times n}{3\times2\times2\times m\times n}$

= $\frac{5}{6}$

 

Question 2: Solve $\frac{y}{x^2 - 25}$ * $\frac{x - 5}{x}$
Solution:
 
Given $\frac{y}{x^2 - 25}$ * $\frac{x - 5}{x}$

Step 1:
$\frac{y}{x^2 - 25}$ * $\frac{x - 5}{x}$ = $\frac{y}{(x - 5)(x + 5)}$ * $\frac{x - 5}{x}$

[ Using identity x2 - y2 = (x - y)(x + y)

⇒ x2 - 25 = (x - 5)(x + 5) ]

Step 2: Cancel the factors common to the numerator and denominator

$\frac{y}{x^2 - 25}$ * $\frac{x - 5}{x}$ = $\frac{y(x - 5)}{x(x - 5)(x + 5)}$

= $\frac{y}{x(x + 5)}$

Step 3:
$\frac{y}{x^2 - 25}$ * $\frac{x - 5}{x}$ = $\frac{y}{x^2 + 5x}$


 

Question 3: Multiply and simplify $\frac{6x^2 + 7x - 3}{x - 1}$ and $\frac{x^2 - 1}{2x^2 + 3x}$
Solution:
 
Given $\frac{6x^2 + 7x - 3}{x - 1}$ * $\frac{x^2 - 1}{2x^2 + 3x}$

Step 1: Factorize the numerators and denominators

6x2 + 7x - 3 = (2x + 3)(3x - 1),

x2 - 1 = (x - 1)(x + 1) and

2x2 + 3x = x(2x + 3)

Step 2:
$\frac{6x^2 + 7x - 3}{x - 1}$ * $\frac{x^2 - 1}{2x^2 + 3x}$

=
$\frac{(2x + 3)(3x - 1)}{x - 1}$ * $\frac{(x - 1)(x + 1)}{x(2x + 3)}$

= $\frac{(x + 1)(3x - 1)}{x}$

Step 3:

$\frac{6x^2 + 7x - 3}{x - 1}$ * $\frac{x^2 - 1}{2x^2 + 3x}$ = $\frac{3x^2 + 2x - 1}{x}$.


 


Dividing Algebraic Fractions

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To divide algebraic fractions, invert the fraction and multiply. We can cancel only after we invert. Division also describe as to divide by a fraction, multiply by its reciprocal.

If $\frac{x}{y}$ and $\frac{m}{n}$ are two rational expressions with y $\neq$ 0 and n $\neq$ 0,

then $\frac{x}{y}$ $\div$ $\frac{m}{n}$ = $\frac{x}{y}$ * $\frac{n}{m}$ = $\frac{x \times n}{y \times m}$ = $\frac{xn}{ym}$.

Steps for Dividing Algebraic Fractions:

Step 1: Invert the second fraction and change the division sign ($\div$) to a multiplication sign ($\times$).

Step 2: Factorize the numerators and denominators.

Step 3: Cancel the factors common to both the numerator and denominator.

Step 4: Multiply the remaining numerators together and denominators together.

Solved Examples

Question 1: Solve $\frac{2x}{x + 1}$ $\div$ $\frac{x + 3}{x + 1}$
Solution:
 
$\frac{2x}{x + 1}$ $\div$ $\frac{x + 3}{x + 1}$ = $\frac{2x}{x + 1}$ * $\frac{x + 1}{x + 3}$

=
$\frac{2x\times (x + 1)}{(x + 1)\times (x + 3)}$

=
$\frac{2x}{x + 3}$
 

Question 2: Divide and simplify $\frac{2}{x^2 + 7x + 10}$ and $\frac{4}{x^2 - 4}$
Solution:
 
Given $\frac{2}{x^2 + 7x + 10}$ $\div$ $\frac{4}{x^2 - 4}$

Step 1:
$\frac{2}{x^2 + 7x + 10}$ $\div$ $\frac{4}{x^2 - 4}$ = $\frac{2}{x^2 + 7x + 10}$ * $\frac{x^2 - 4}{4}$

Step 2: Factorize the numerators and denominators

x2 + 7x + 10 = (x + 2)(x + 5) and

x2 - 4 = (x - 2)(x + 2)

Step 3:
$\frac{2}{x^2 + 7x + 10}$ * $\frac{x^2 - 4}{4}$ = $\frac{2}{(x + 2)(x + 5)}$ * $\frac{(x - 2)(x + 2)}{2\times2}$ = $\frac{x - 2}{x - 5}$

$\frac{2}{x^2 + 7x + 10}$ $\div$ $\frac{4}{x^2 - 4}$ = $\frac{x - 2}{x - 5}$
 

Question 3: Solve $\frac{x^2 - 36}{x^2 + 5x}$ $\div$ $\frac{x - 6}{x + 5}$
Solution:
 
Step 1:
$\frac{x^2 - 36}{x^2 + 5x}$ $\div$ $\frac{x - 6}{x + 5}$ = $\frac{x^2 - 36}{x^2 + 5x}$ * $\frac{x + 5}{x - 6}$


Step 2:
Factorize the numerators and denominators

x2 - 36 = (x - 6)(x + 6)

x2  + 5x = x(x + 5)

Step 3:
$\frac{x^2 - 36}{x^2 + 5x}$ * $\frac{x + 5}{x - 6}$ = $\frac{(x - 6)(x + 6)}{x(x + 5)}$ * $\frac{x + 5}{x - 6}$

=
$\frac{x + 6}{x}$

$\frac{x^2 - 81}{x^2 + 5x}$ $\div$ $\frac{x - 9}{x + 5}$ = $\frac{x + 9}{x}$.
 

Adding Algebraic Fractions

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The following steps can be used for the addition of algebraic fractions, including fractions having common denominators and fractions having different denominators.

Steps For Adding Algebraic Fractions:

If algebraic fractions having same denominators

Step 1: Keep the denominator as it is and combine the numerators.

Step 2: Reduce the fraction if possible.

Steps For Adding Algebraic Fractions:

If algebraic fractions having different denominators

Step 1:
Find a lowest common denominator.

Step 2:
Change each fraction to an equivalent fraction with the common denominator.

Step 3:
Combine numerators and reduce if possible.

Solved Examples

Question 1: Add $\frac{x + 2}{x + 3}$ and $\frac{x + 1}{x + 3}$

Solution:
 
Given $\frac{x + 2}{x + 3}$ + $\frac{x + 1}{x + 3}$

Algebraic expression having same denominator

$\frac{x + 2}{x + 3}$ + $\frac{x + 1}{x + 3}$ = $\frac{x + 2 + x + 1}{x + 3}$

= $\frac{2x + 3}{x + 3}$

$\frac{x + 2}{x + 3}$ + $\frac{x + 1}{x + 3}$ = $\frac{2x + 3}{x + 3}$.
 

Question 2: Express $\frac{1}{x + 5}$ + $\frac{x}{x - 3}$ as a single fraction.
Solution:
 
Given $\frac{1}{x + 5}$ + $\frac{x}{x - 3}$

Step 1:

Lowest common denominator of (x + 5) and (x - 3) is (x + 5)(x - 3)

Step 2: Multiply and divide $\frac{1}{x + 5}$  by x - 3

and $\frac{x}{x - 3}$ by x + 5, we have

$\frac{1}{x + 5}$ = $\frac{x - 3}{(x + 5)(x - 3)}$

and

$\frac{x}{x - 3}$ = $\frac{x(x + 5)}{(x - 3)(x + 5)}$

Step 3:

$\frac{1}{x + 5}$ + $\frac{x}{x - 3}$ = $\frac{x - 3}{(x + 5)(x - 3)}$ + $\frac{x(x + 5)}{(x - 3)(x + 5)}$

= $\frac{x - 3 + x(x + 5)}{(x + 5)(x - 3)}$

= $\frac{x - 3 + x^2 + 5x}{(x + 5)(x - 3)}$

= $\frac{x^2 + 6x - 3}{(x + 5)(x - 3)}$.
 

Question 3: Solve $\frac{x + 3}{x + 1}$ + $\frac{6}{x - 2}$
Solution:
 
Given $\frac{x + 3}{x + 1}$ + $\frac{6}{x - 2}$

Step 1: Lowest common denominator of (x + 1) and (x - 2) is (x + 1)(x - 2).

Step 2:

$\frac{x + 3}{x + 1}$ + $\frac{6}{x - 2}$ = $\frac{(x + 3)(x - 2) + 6(x+1)}{(x + 1)(x - 2)}$


= $\frac{x^2 - 2x + 3x - 6 + 6x + 6}{x^2 - x - 2}$

= $\frac{x^2 + 7x}{x^2 - x - 2}$

$\frac{x + 3}{x + 1}$ + $\frac{6}{x - 2}$ = $\frac{x^2 + 7x}{x^2 - x - 2}$.
 

Subtracting Algebraic Fractions

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To subtract algebraic fractions in the same way that we do in simple arithmetic. The following steps can be used for the subtraction of algebraic fractions, including fractions having common denominators and fractions having different denominators.
Steps For Subtracting Algebraic Fractions:

If algebraic fractions having same denominators

Step 1: Keep the denominator as it is and subtract the numerators.

Step 2: Reduce the fraction if possible.
Steps For Subtracting Algebraic Fractions:

If algebraic fractions having different denominators

Step 1:
Find a lowest common denominator.

Step 2:
Change each fraction to an equivalent fraction with the common denominator.

Step 3:
Subtract numerators and reduce if possible.

Solved Examples

Question 1: Express $\frac{2x + 1}{x + 5}$ - $\frac{x}{x + 5}$ as a single fraction.

Solution:
 
Given $\frac{2x + 1}{x + 5}$ - $\frac{x}{x + 5}$

Step 1:

Both the fractions have common denominator = (x + 5)

Step 2:  Keep the denominator as it is and add the numerators

$\frac{2x + 1}{x + 5}$ - $\frac{x}{x + 5}$  = $\frac{2x + 1 - x}{x + 5}$

= $\frac{2x - x + 1}{x + 5}$

= $\frac{x + 1}{x + 5}$
 

Question 2: Express $\frac{2}{x + 1}$ - $\frac{x}{3}$ as a single fraction.
Solution:
 
Given $\frac{2}{x + 1}$ - $\frac{x}{3}$

Step 1:

Lowest common denominator of (x + 1) and 3 is 3(x + 1)

Step 2: Multiply and divide $\frac{2}{x + 1}$ by  3

and $\frac{x}{3}$ by x + 1, we have

$\frac{2}{x + 1}$ = $\frac{2\times3}{(x + 1)\times3}$ = $\frac{6}{(x + 1)\times3}$

and

$\frac{x}{3}$ = $\frac{x(x + 1)}{3(x + 1)}$

Step 3:

$\frac{2}{x + 1}$ - $\frac{x}{3}$  = $\frac{6}{(x + 1)\times3}$ - $\frac{x(x + 1)}{3(x + 1)}$

=
$\frac{6 - x^2 - x}{3(x + 1)}$

=
$\frac{- x^2 - x + 6}{3(x + 1)}$

 

Question 3: Solve $\frac{2x}{x + 1}$ - $\frac{x}{x + 5}$
Solution:
 
Given $\frac{2x}{x + 1}$ - $\frac{x}{x + 5}$

Step 1: Lowest common denominator of (x + 1) and (x + 5) is (x + 1)(x + 5).

Step 2:
$\frac{2x}{x + 1}$ - $\frac{x}{x + 5}$ = $\frac{2x(x + 5) - x(x + 1)}{(x + 1)(x + 5)}$

= $\frac{2x^2 + 10x - x^2 - x}{(x + 1)(x + 5)}$

= $\frac{x^2 + 9x}{(x + 1)(x + 5)}$
 

Question 4: Solve $\frac{y}{2}$ - $\frac{x}{y}$
Solution:
 
Given $\frac{y}{2}$ - $\frac{x}{y}$

Step 1: Lowest common denominator of 2 and y is 2y.

Step 2:
$\frac{y}{2}$ - $\frac{x}{y}$ = $\frac{y^2 - 2x}{2y}$

 

Question 5: Solve $\frac{m + 1}{m - 2}$ - $\frac{1}{m}$
Solution:
 
Given $\frac{m + 1}{m - 2}$ - $\frac{1}{m}$

Step 1: Lowest common denominator of m - 2 and m is m(m - 2).

Step 2:
$\frac{m + 1}{m - 2}$ - $\frac{1}{m}$ = $\frac{m(m + 1) - (m - 2)}{m(m - 2)}$

=
$\frac{m^2 + m - m + 2}{m(m - 2)}$

=
$\frac{m^2 + 2}{m(m - 2)}$

 

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