Q16. - Algebra 1

Question

Find the product.

$\frac{\mathbf{m}}{{\mathbf{m}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{m}\mathbf{+}\mathbf{2}}\mathbf{.}\frac{\mathbf{m}\mathbf{+}\mathbf{2}}{{\mathbf{m}}^{\mathbf{2}}}$ 

Step-by-Step Solution

Answer

The final answer is $\frac{\mathbf{1}}{\mathbf{m}\mathbf{(}\mathbf{m}\mathbf{+}\mathbf{1}\mathbf{)}}$.

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 product of two expressions a b and c d .

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

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

Step 3. Calculate the product m m 2 + 3 m + 2 . m + 2 m 2 .

Use $\left(1\right)$ to find the product $\frac{m}{m^2+3m+2}.\frac{m+2}{m^2}$.

$\begin{array}{c}\frac{m}{m^2+3m+2}.\frac{m+2}{m^2}=\frac{m(m+2)}{(m^2+3m+2)\left(m^2\right)}\\ =\frac{m+2}{m(m^2+3m+2)}\\ =\frac{m+2}{m(m^2+2m+m+2)}\\ =\frac{m+2}{m(m(m+2)+1(m+2))}\\ =\frac{m+2}{m(m+2)(m+1)}\\ =\frac{1}{m(m+1)}\end{array}$

Therefore, the final answer is $\frac{1}{m(m+1)}$.