#### Answer

$6x^{5} +3 - \frac{1}{x}$

#### Work Step by Step

Given $\frac{(24x^{7} + 12x^{2} – 4x)}{4x^{2}}$
the answer, simplified, is $6x^{5} +3 - \frac{1}{x}$
• We can separate the polynomial (top) by its terms to divide it by the monomial (bottom). So, if we separate the polynomial (the quotient, which is also the numerator of the equation), it looks like this:
Term #1: $24x^{7}$
Term #2: $12x^{2}$
Term #3: $-4x$
Adding its divisor, our equation would then look like this:
$\frac{24x^{7}}{4x^{2}}$ $+$ $\frac{12x^{2}}{4x^{2}}$ $-$ $\frac{4x}{4x^{2}}$
• Now we need to simplify each fraction, consisting of the one term of the polynomial and the monomial. And we can further simplify each term into its coefficients and exponents to make it a little easier. So, our equation will then look like this:
$(\frac{24}{4})$$(\frac{x^{7}}{ x^{2}})$ $+$ $(\frac{12}{4})$$(\frac{x^{2}}{ x^{2}})$ $-$ $(\frac{4}{4})$$(\frac{x}{ x^{2}})$
• Now, let's take the 1st term that we have set up, and divide the coefficients and the exponents. (Remember that when we divide exponents, we are really just subtracting the denominator's exponent from the numerator's exponent.)
For $(\frac{24}{4})$$(\frac{x^{7}}{ x^{2}})$,
$(\frac{24}{4})$ $=6$ and $(\frac{x^{7}}{ x^{2}})$ is $x^{7-2}$ $=x^{5}$
• Now, let's take the 2nd term that we have set up, and divide the coefficients and the exponents. (Remember that when we divide exponents, we are really just subtracting the denominator's exponent from the numerator's exponent.)
For $(\frac{12}{4})$$(\frac{x^{2}}{ x^{2}})$,
$(\frac{12}{4})$ $=3$ and $(\frac{x^{2}}{ x^{2}})$ is $x^{2-2}$ $=x^{0}$, simplified to $1$, because anything to the 0 power is 1.
• Now, let's take the 3rd term that we have set up, and divide the coefficients and the exponents. (Remember that when we divide exponents, we are really just subtracting the denominator's exponent from the numerator's exponent.)
For $-$$(\frac{4}{4})$$(\frac{x}{ x^{2}})$.
$-$$(\frac{4}{4})$ $=-1$ and $(\frac{x}{x^{2}})$ is $x^{1-2}$ $=x^{-1}$ However:
Because the base's exponent is negative, we need to find its reciprocal in order to simplify this fraction. So:
$x^{-1}$ $=$ $(\frac{1}{x})$
• Now that we have simplified each polynomial's terms when each is divided by the monomial, our equation of
$(\frac{24}{4})$$(\frac{x^{7}}{ x^{2}})$ $+$ $(\frac{12}{4})$$(\frac{x^{2}}{ x^{2}})$ $-$ $(\frac{4}{4})$$(\frac{x}{ x^{2}})$
may be simplified to the following based on our steps:
$(6)(x^{5})$ $+ (3)(1)$ $-(1)$$(\frac{1}{x})$,
which is then further simplified to the final answer of:
$6x^{5} +3 - \frac{1}{x}$