Question

Please help me! Thank you!

Given f(x)=4 x^2+6 x and g(x)=2 x^2+13 x+15, find (f/g)(x)
Show your work.

Answer 1

Answer:

To find (f/g)(x), we need to divide the function f(x) by the function g(x). The notation (f/g)(x) represents the quotient of f(x) divided by g(x).

f(x)=4x²+6x

g(x)=2x²+13x+15

To divide f(x) by g(x), we perform polynomial long division or use the method of partial fractions. Let's use polynomial long division:

2x & 2x + 7 \\

\hline

4x^2 + 6x & 4x^2 + 13x + 15 \\

& -(4x^2 + 26x) \\

& \underline{\phantom{-(4x^2 + 26x + 30)}} \\

& -13x + 15 \\

& \underline{\phantom{-(-13x - 21)}} \\

& 36

\end{array}

So, the quotient of f(x) divided by g(x) is 2x+7.

Therefore,

(f/g)(x)=2x+7.

Answer 2

Answer:

$\displaystyle\left(\frac{f}{g} \right)(x)=\frac{2x}{x+5}$

Step-by-step explanation:

To simplify a Polynomial Division, factorize the polynomial function and see if there is any common factor between numerator and denominator.

$\displaystyle \left(\frac{f}{g} \right)(x)=\frac{4x^2+6x}{2x^2+13x+15}$

             $\displaystyle=\frac{2x(2x+3)}{(x+5)(2x+3)}$

We can eliminate the common factor (2x+3) for both numerator and denominator, therefore:

             $\displaystyle=\frac{2x}{x+5}$