Multiplying polynomial fractionsPolynomial 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)

eq 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 Numeratorleftarrow DenominatorThe principle which we apply while multiplying two fraction i.e. frac{a}{b}*frac{c}{d} = frac{a*c}{b*d} where b

eq 0 and d

eq 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}{12} = frac{1}{3}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 fractionsFactor 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} = -4Note: 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}/latexExample 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}ExerciseMultiply 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}