Q20. - Algebra 1

Question

Find the quotient.

$\frac{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{7}\mathbf{x}\mathbf{+}\mathbf{12}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{25}}\mathbf{÷}\frac{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{9}}{\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{10}}$

Step-by-Step Solution

Answer

The final answer is $\frac{\mathbf{2}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{4}\mathbf{)}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{5}\mathbf{)}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{3}\mathbf{)}}$.

Step 1. Define a rational function.

A function of the form $y=\frac{p}{q}$ is called a rational function, where $p$ and $q$ are the polynomials and $q\ne 0$.

Step 2. Define the quotient of two expressions a b and c d .

The quotient of two expressions $\frac{\mathrm{a}}{\mathrm{b}}$ and $\frac{\mathrm{c}}{\mathrm{d}}$, where $b\ne 0$ and $d\ne 0$, is given below.

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=\frac{ac}{bd}\mathrm{...........}\left(1\right)$

Step 3. Calculate the quotient x 2 + 7 x + 12 x 2 − 25 ÷ x 2 − 9 2 x + 10 .

Use $\left(1\right)$ to find the quotient $\frac{x^2+7x+12}{x^2-25}÷\frac{x^2-9}{2x+10}$.

$\begin{array}{r}\frac{x^2+7x+12}{x^2-25}÷\frac{x^2-9}{2x+10}=\left(\frac{x^2+7x+12}{x^2-25}\right)\times \left(\frac{2x+10}{x^2-9}\right)\\ =\frac{(x^2+7x+12)(2x+10)}{(x^2-25)(x^2-9)}\\ =\frac{(x^2+4x+3x+12)\left[2(x+5)\right]}{(x^2-5^2)(x^2-9)}\\ =\frac{\left[x(x+4)+3(x+4)\right]\left[2(x+5)\right]}{(x+5)(x-5)(x+3)(x-3)}\\ =\frac{(x+4)(x+3)\left(2\right)(x+5)}{(x+5)(x-5)(x+3)(x-3)}\\ =\frac{2(x+4)}{(x-5)(x-3)}\end{array}$

Therefore, the final answer is $\frac{2(x+4)}{(x-5)(x-3)}$.