Subtraction of Polynomial Fractions

Polynomial fraction is in the form of the ratio of two polynomials like $\frac{P(x)}{Q(x)}$ where divisible of zero is not allowed,like $Q(x)\neq 0$ . Various operations can be performed same as we do in simple arithmetic such as add, divide, multiply and subtract.Polynomial fraction is an expression of a polynomial divided by another polynomial. Let P(x) and Q(x), where Q(x) cannot be zero.

$F(x)=\frac{P(x)}{Q(x)}$ = $\frac{9x-6-18}{x^{2}-x-3}$ $\leftarrow Numerator\\\leftarrow Denominator$

The principle which we apply while subtracting two fraction i.e. $\frac{a}{c}-\frac{b}{c} = \frac{a-b}{c}$ where $c\neq 0$ , the same principle is being applied while subtracting two polynomial fractions containing variables,coeficient in it.

Example: Sub $\frac{8}{9}$ – $\frac{4}{9}$

Solution: By applying the principle of subtracting two fraction we get, $\frac{8}{9}$ – $\frac{4}{9}$ = $\frac{4}{9}$

Subtraction of two expressions with common denominators

Step 1. Subtract both of the numerator

Step 2. Take difference of both the numerators in step 1 and place it over the common denominator.

Step 3. Simplify the fraction further by factorizing if possible.

Example 1: Sub $\frac{10}{x-6}$ – $\frac{x+5}{x-6}$

Solution: Given expression $\frac{10}{x-6}$ – $\frac{x+5}{x-6}$ = $\frac{10-(x+5)}{x-6}$ = $\frac{5-x}{x-6}$

Example 2: Sub $\frac{x^{2}-2x+3}{x^{2}+7x+12}$ – $\frac{x^{2}-4x-5}{x^{2}+7x+12}$

Solution: Given expression $\frac{x^{2}-2x+3}{x^{2}+7x+12}$ – $\frac{x^{2}-4x-5}{x^{2}+7x+12}$ = $\frac{(x^{2}-2x+3)-(x^{2}-4x-5)}{x^{2}+7x+12}$ = $\frac{x^{2}-2x+3-x^{2}+4x+5}{x^{2}+7x+12}$ = $\frac{2x+8}{x^{2}+7x+12}$

Factoring the expression $\frac{2x+8}{x^{2}+7x+12}$ = $\frac{2(x+4)}{(x+3)(x+4)}$ = $\frac{2}{x+3}$

We can subtract polynomial fractions with only common denominator but if we don’t have the common denominator, we have to find the least common denominator i.e. LCD which will give us the smallest expression that is divisible by both the denominators. It is also known as least common multiple.

Steps to find LCD [Least common denominators]

Find the LCM [Least common multiple] of both the expressions.
Change each of the polynomial fractions to make their denominators equal to the LCD.
Subtract both the expressions.

Example: Find the LCD $\frac{2}{x^{2}}$ – $\frac{3}{7x}$

Solution: There are two denominators $x^{2}$ and 7x. So by placing each factor with its highest power we get the LCD $7x^{2}$ .

Subtraction of two expressions with different denominators

Step1. Find the LCD.

Step 2. Change each of the fractions same as the LCD by multiplying the numerators as well as the denominator of each expression by any factors which make it equal to the LCD.

Step 3. Subtract both of the numerators.

Step 4. Simplify the numerator by factoring it, if possible.

Example 1: Subtract $\frac{x}{x+5}$ and $\frac{2}{x-3}$

Solution: The LCD of x+5 and x-3 is (x+5)(x+3)

Multiply the numerators as well as the denominator of each expression by any factors which make it equal to the LCD $\frac{x}{x+5}$ * $\frac{x-3}{x-3}$ – $\frac{2}{x-3}$ * $\frac{(x+5)}{(x+5)}$ = $\frac{x^2-3x}{(x-3)(x+5)}$ – $\frac{2x+10}{(x-3)(x+5)}$

Subtract the numerators $\frac{x^2-3x}{(x-3)(x+5)}$ – $\frac{2x+10}{(x-3)(x+5)}$ = $\frac{x^2-3x-2x-10}{(x-3)(x+5)}$ = $\frac{x^2-5x-10}{(x-3)(x+5)}$

Example 2: Subtract $\frac{5}{x^2-5x}$ and $\frac{x}{5x-25}$

Solution: The LCD of x^2-5x and 5x-25 is 5x(x-5)

Multiply the numerators as well as the denominator of each expression by any factors which make it equal to the LCD $\frac{5}{x^2-5x}$ * $\frac{5}{5}$ – $\frac{x}{5x-25}$ * $\frac{(x)}{(x)}$ = $\frac{25}{5x(x-5)}$ – $\frac{x^2}{5x(x-5)}$