Operations with algebraic fractions

Example

Compute the sum of the following algebraic fractions ${\displaystyle \frac{x-1}{x+4}}\text{and}{\displaystyle \frac{x^2+2}{x+4}}$

In this case, both fractions have the same denominator, and we can then compute straight away: $${\displaystyle \frac{x-1}{x+4}}+{\displaystyle \frac{x^2+2}{x+4}}={\displaystyle \frac{x-1+(x^2+2)}{x+4}}={\displaystyle \frac{x^2+x+1}{x+4}}$$

Example

Compute the subtraction of the following algebraic fractions ${\displaystyle \frac{x^2+1}{x-2}}\text{and}{\displaystyle \frac{x+1}{x-1}}$

First, we have to convert the algebraic fractions into fractions with common denominator:

$lcm\{x-2,x-1\}=(x-2)\cdot (x-1)$

${\displaystyle \frac{(x-2)\cdot (x-1)}{(x-2)}}=x-1\Rightarrow (x-1)\cdot (x^2+1)=x\cdot (x^2+1)-1\cdot (x^2+1)=$

$=x^3-x^2+x-1\Rightarrow {\displaystyle \frac{x^3-x^2+x-1}{(x-2)\cdot (x-1)}}$

${\displaystyle \frac{(x-2)\cdot (x-1)}{(x-1)}}=x-2\Rightarrow (x-1)\cdot (x+1)=x^2-1\Rightarrow {\displaystyle \frac{x^2-1}{(x-2)\cdot (x-1)}}$

Now we compute:

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

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

Example

Compute the subtraction of the following algebraic fractions ${\displaystyle \frac{x-2}{x+3}}\text{and}{\displaystyle \frac{x-1}{(x+1{)}^2}}$

First, we have to convert the algebraic fractions into fractions with common denominator:

$lcm\{x+3,(x+1{)}^2\}=(x+3)\cdot (x+1{)}^2$

${\displaystyle \frac{(x+3)\cdot (x+1{)}^2}{x+3}}=(x+1{)}^2\Rightarrow (x-2)\cdot (x+1{)}^2=x\cdot (x+1{)}^2+1\cdot (x+1{)}^2=$

$=x\cdot (x^2+2x+1)+1\cdot (x^2+2x+1)=x^3+3x^2+3x+1\Rightarrow$

$\Rightarrow {\displaystyle \frac{x^3+3x^2+3x+1}{(x+3)\cdot (x+1{)}^2}}$

${\displaystyle \frac{(x+3)\cdot (x+1{)}^2}{(x+1{)}^2}}=x+3\Rightarrow (x-1)\cdot (x+3)=x^2+2x-3\Rightarrow$

$\Rightarrow {\displaystyle \frac{x^2+2x-3}{(x+3)\cdot (x+1{)}^2}}$

Now we compute:

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

$={\displaystyle \frac{x^3+2x^2-x+4}{(x+3)(x+1{)}^2}}$

Example

Compute the product of the following algebraic fractions ${\displaystyle \frac{x-1}{x+4}}$ and ${\displaystyle \frac{x^2+2}{x-2}}$.

We multiply numerators and denominators, and obtain the desired result:

${\displaystyle \frac{x-1}{x+4}}\cdot {\displaystyle \frac{x^2+2}{x-2}}={\displaystyle \frac{(x-1)\cdot (x^2+2)}{(x+4)\cdot (x-2)}}={\displaystyle \frac{x\cdot (x^2+2)-1\cdot (x^2+2)}{x\cdot (x-2)+4\cdot (x-2)}}=$

$={\displaystyle \frac{x^3-x^2+2x-2}{x^2+2x-8}}$

Example

Compute the product of the following algebraic fractions ${\displaystyle \frac{x+5}{x}}$ and ${\displaystyle \frac{x^2-1}{x+3}}$.

We multiply numerators and denominators, and obtain the desired result:

${\displaystyle \frac{x+5}{x}}\cdot {\displaystyle \frac{x^2-1}{x+3}}={\displaystyle \frac{(x+5)\cdot (x^2-1)}{x\cdot (x+3)}}={\displaystyle \frac{x\cdot (x^2-1)+5\cdot (x^2-1)}{x\cdot (x+3)}}=$

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

Example

Compute the division of the following algebraic fractions ${\displaystyle \frac{x-1}{x+4}}$ and ${\displaystyle \frac{x^2+2}{x-2}}$.

We multiply the first fraction by the second inverted, and obtain the desired result:

${\displaystyle \frac{x-1}{x+4}}\cdot {\displaystyle \frac{x-2}{x^2+2}}={\displaystyle \frac{(x-1)\cdot (x-2)}{(x+4)\cdot (x^2+2)}}={\displaystyle \frac{x\cdot (x-2)-1\cdot (x-2)}{x\cdot (x^2+2)+4\cdot (x^2+2)}}=$

$={\displaystyle \frac{x^2-3x+2}{x^3+4x^2+2x+8}}$

Example

Compute the division of the following algebraic fractions ${\displaystyle \frac{x+5}{x}}$ and ${\displaystyle \frac{x^2-1}{x+3}}$.

We multiply the first fraction by the second inverted, and obtain the desired result:

${\displaystyle \frac{x+5}{x}}\cdot {\displaystyle \frac{x+3}{x^2-1}}={\displaystyle \frac{(x+5)\cdot (x+3)}{x\cdot (x^2-1)}}={\displaystyle \frac{x\cdot (x+3)+5\cdot (x+3)}{x\cdot (x^2-1)}}=$

$={\displaystyle \frac{x^2+8x+15}{x^3-x}}$