Multiplying 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.

$F(x)=\frac{P(x)}{Q(x)}$ = $\frac{10x-5-20}{x^{2}-x-9}$ $\leftarrow Numerator\\\leftarrow Denominator$

The principle which we apply while multiplying two fraction i.e. $\frac{a}{b}*\frac{c}{d} = \frac{a*c}{b*d}$ where $b\neq 0$ and $d\neq 0$ , the same principle is being applied while multiplying two polynomial fractions containing variables and coeficient in it. To multiply polynomial, first to factor both the numerator and denominator of both the expressions and then multiply the remaining polynomial.

Example 1: Multiply $\frac{3}{8}$ and $\frac{4}{9}$

Solution: Divide out any common factors to both a numerator and denominator and then multiply them:

$\frac{3}{8}$ * $\frac{4}{9}$ = $\frac{1*4}{8*3}$ = $\frac{4}{24}$ = $\frac{1}{6}$

Example 2: Multiply $\frac{7x^2}{3}$ and $\frac{9}{14x}$

Solution: Given expression $\frac{7x^2}{3}$ * $\frac{9}{14x}$

By dividing out any common factors to both a numerator and denominator and then multiply them we get: $\frac{x}{1}$ * $\frac{3}{2}$ = $\frac{3x}{2}$

Steps to multiply the polynomial fractions

Factor each the numerators and denominators of all fractions completely.
Cancel or reduce the fractions. keep in mind that to reduce fractions; you’ll be able to cancel something within the numerator with one thing within the denominator, however, so as to cancel something within the numerator and denominator the 2 factors should be precisely the same.
Rewrite the remaining factor. Notice that you simply don’t need to really to multiply something within numerator or denominator.

Example 1: Multiply $\frac{3x+2}{2x+1}$ and $\frac{4-8x}{3x+2}$

Solution: 1. By factoring completely the numerator and denominator,if possible we get $\frac{3x+2}{2x+1}$ * $\frac{4-8x}{3x+2}$ = $\frac{3x+2}{2x+1}$ * $\frac{4(1-2x)}{3x+2}$

2. Cancel the common terms which are same in both numerator and denominator: $\frac{3x+2}{2x+1}$ * $\frac{4(1-2x)}{3x+2}$ = $\frac{3x+2}{2x+1}$ * $\frac{4(-1)(2x-1)}{3x+2}$

3. Rewrite the remaining factor: $\frac{-4}{1}$ = -4

Note: When multiplying polynomial expression and if there is a sign differ in both a numerator and denominator. For example the numerator is x-2 and the denominator 2-x by factoring out -1 from the numerator or denominator and then divide out the common factors.

Example 2: Multiply $\frac{x^2y+2y^2}{x^2-1}$ and $\frac{x+1}{x^2+2y}$

Solution: 1. By factoring completely the numerator and denominator,if possible we get $\frac{x^2y+2y^2}{x^2-1}$ * $\frac{x+1}{x^2+2y}$ = $\frac{y(x^2+2y)}{(x-1)(x+1)}$ * $\frac{x+1}{x^2+2y}$

2. Cancel the common terms which are same in both numerator and denominator: $\frac{y(x^2+2y)}{(x-1)(x+1)}$ * $\frac{x+1}{x^2+2y}$ = latex] \frac{y}{x-1}[/latex]

Example 3: Multiply $\frac{12x-4x^2}{x^2+x-12}$ and $\frac{x^2+2x-8}{x^3-4x}$

Solution: 1. By factoring completely the numerator and denominator,if possible we get $\frac{12x-4x^2}{x^2+x-12}$ * $\frac{x^2+2x-8}{x^3-4x}$ = $\frac{-4x(x-3)}{(x-3)(x+4)}$ * $\frac{(x-2)(x+4)}{x(x+2)(x-2)}$

2. Cancel the common terms which are same in both numerator and denominator and rewrite the fraction: $\frac{-4x(x-3)}{(x-3)(x+4)}$ * $\frac{(x-2)(x+4)}{x(x+2)(x-2)}$ = $\frac{-4}{x+2}$

Exercise

Multiply the following polynomial fractions

$\frac{3y^2}{5}$ and $\frac{10x}{15y}$
$\frac{5y^2}{3}$ and $\frac{9x}{10y}$
$\frac{9y^2}{8}$ and $\frac{32x}{27y}$
$\frac{xy+2y}{x^2-100}$ and $\frac{x-10}{xy-y}$
$\frac{x^2-16}{x^2}$ and $\frac{x^2-4x}{x^2-x-12}$
$\frac{x^2-2x-35}{2x^3-3x^2}$ and $\frac{4x^3-9x}{7x-49}$

Comments

rico says

September 20, 2020 at 6:15 pm

\frac{3}{8} * \frac{4}{9} = \frac{1*4}{8*3} = \frac{4}{12} =

Please correct the above from 4/12 to 4 /24 as 8×3 = 24 and further reduced then to 1/6

- Maths Tutor says
  
  September 24, 2020 at 3:03 am
  
  Thanks for pointing that out and it is now updated.

Steps to multiply the polynomial fractions

Exercise

Comments

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