Question

a rational expression is a fraction formed by a polynomial numerator and a polynomial denominator. it is in its simplest form when the only common factor between the numerator and the denominator is .
\\(\frac{3x^2}{x^2 + 3x}\\) \\(\frac{3a^2 - 15a}{3a^2 - 75}\\) \\(\frac{m^2 - 4m + 3}{(m - 3)(m + 1)}\\) \\(\frac{p^2 - p - 12}{p^2 - 6p + 8}\\)
multiplying rational expressions is done by “extending” the “fraction bar” (vinculum) to perform multiplication between the numerators and between the denominators.
\\(\frac{2n^2 - 8}{n^2 - 2n} \cdot \frac{3n}{2n^2 + 4n}\\) \\(\frac{5y + 10}{5y} \cdot \frac{y^2 + y}{y^2 - 4}\\) \\(\frac{6r}{8r^2 - 50} \cdot \frac{2r^2 + 3r - 5}{3r - 3}\\)
\\(\frac{x^2}{x^2 - 1} \cdot \frac{x^2 + 3x + 2}{x^2 + 2x}\\) \\(\frac{a^2 - 2ab + b^2}{10ab} \cdot \frac{5ab}{b^2 - a^2}\\) \\(\frac{6xy}{(x - y)^2} \cdot \frac{y - x}{2x + 2y}\\)

Explanation:

Part 1: Simplify each rational expression

Step1: Factor numerator & denominator

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

Step2: Cancel common factor $x$

$\frac{3x \cdot \cancel{x}}{\cancel{x}(x+3)} = \frac{3x}{x+3}$

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Step1: Factor numerator & denominator

$\frac{3a^2-15a}{3a^2-75} = \frac{3a(a-5)}{3(a^2-25)} = \frac{3a(a-5)}{3(a-5)(a+5)}$

Step2: Cancel common factors $3, (a-5)$

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

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Step1: Factor numerator & denominator

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

Step2: Cancel common factor $(m-3)$

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

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Step1: Factor numerator & denominator

$\frac{p^2-p-12}{p^2-6p+8} = \frac{(p-4)(p+3)}{(p-4)(p-2)}$

Step2: Cancel common factor $(p-4)$

$\frac{\cancel{(p-4)}(p+3)}{\cancel{(p-4)}(p-2)} = \frac{p+3}{p-2}$

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Part 2: Multiply and simplify rational expressions

Step1: Factor all terms

$\frac{2n^2-8}{n^2-2n} \cdot \frac{3n}{2n^2+4n} = \frac{2(n^2-4)}{n(n-2)} \cdot \frac{3n}{2n(n+2)} = \frac{2(n-2)(n+2)}{n(n-2)} \cdot \frac{3n}{2n(n+2)}$

Step2: Cancel common factors

$\frac{\cancel{2}\cancel{(n-2)}\cancel{(n+2)}}{\cancel{n}\cancel{(n-2)}} \cdot \frac{3\cancel{n}}{\cancel{2}n\cancel{(n+2)}} = \frac{3}{n}$

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Step1: Factor all terms

$\frac{5y+10}{5y} \cdot \frac{y^2+y}{y^2-4} = \frac{5(y+2)}{5y} \cdot \frac{y(y+1)}{(y-2)(y+2)}$

Step2: Cancel common factors

$\frac{\cancel{5}\cancel{(y+2)}}{\cancel{5}\cancel{y}} \cdot \frac{\cancel{y}(y+1)}{(y-2)\cancel{(y+2)}} = \frac{y+1}{y-2}$

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Step1: Factor all terms

$\frac{6r}{8r^2-50} \cdot \frac{2r^2+3r-5}{3r-3} = \frac{6r}{2(4r^2-25)} \cdot \frac{(2r+5)(r-1)}{3(r-1)} = \frac{6r}{2(2r-5)(2r+5)} \cdot \frac{(2r+5)(r-1)}{3(r-1)}$

Step2: Cancel common factors

$\frac{\cancel{6}r}{\cancel{2}(2r-5)\cancel{(2r+5)}} \cdot \frac{\cancel{(2r+5)}\cancel{(r-1)}}{\cancel{3}\cancel{(r-1)}} = \frac{r}{2r-5}$

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Step1: Factor all terms

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

Step2: Cancel common factors

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

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Step1: Factor all terms

$\frac{a^2-2ab+b^2}{10ab} \cdot \frac{5ab}{b^2-a^2} = \frac{(a-b)^2}{10ab} \cdot \frac{5ab}{(b-a)(b+a)} = \frac{(a-b)^2}{10ab} \cdot \frac{5ab}{-(a-b)(a+b)}$

Step2: Cancel common factors

$\frac{\cancel{(a-b)^2}}{\cancel{10}\cancel{ab}} \cdot \frac{\cancel{5}\cancel{ab}}{-\cancel{(a-b)}(a+b)} = -\frac{a-b}{2(a+b)} = \frac{b-a}{2(a+b)}$

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Step1: Factor all terms

$\frac{6xy}{(x-y)^2} \cdot \frac{y-x}{2x+2y} = \frac{6xy}{(x-y)^2} \cdot \frac{-(x-y)}{2(x+y)}$

Step2: Cancel common factors

$\frac{\cancel{6}xy}{(x-y)^{\cancel{2}}} \cdot \frac{-\cancel{(x-y)}}{\cancel{2}(x+y)} = -\frac{3xy}{(x-y)(x+y)}$

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Answer:

s:

Simplified Rational Expressions:

1. $\boldsymbol{\frac{3x}{x+3}}$
2. $\boldsymbol{\frac{a}{a+5}}$
3. $\boldsymbol{\frac{m-1}{m+1}}$
4. $\boldsymbol{\frac{p+3}{p-2}}$

Multiplied & Simplified Expressions:

1. $\boldsymbol{\frac{3}{n}}$
2. $\boldsymbol{\frac{y+1}{y-2}}$
3. $\boldsymbol{\frac{r}{2r-5}}$
4. $\boldsymbol{\frac{x}{x-1}}$
5. $\boldsymbol{\frac{b-a}{2(a+b)}}$
6. $\boldsymbol{-\frac{3xy}{(x-y)(x+y)}}$