#### Answer

Given $\frac{(20x^{3} - 30x^{2} +5x + 5)}{5}$
the answer, simplified, is $4x^{3} - 6x^{2} + x + 1$

#### Work Step by Step

Given $\frac{(20x^{3} - 30x^{2} +5x + 5)}{5}$
the answer, simplified, is $4x^{3} - 6x^{2} + x + 1$
• 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: $20x^{3}$
Term #2: $- 30x^{2}$
Term #3: $5x$
Term #4: $5$
Adding its divisor, our equation would then look like this:
$\frac{20x^{3}}{5}$ $-$ $\frac{30x^{2}}{5}$ $+$ $\frac{5x}{5}$ $+$ $\frac{5}{5}$
• 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. Since our divisor is a Constant, however, our fractions with the exponents can be written either of the following ways because only similar terms may be divided by similar terms:
$(\frac{Polynomial Term}{5x^{0}})$ OR $(\frac{Polynomial Term}{(5)(1)})$,
Remember that anything raised to the 0 power = 1. So, our equation can look either way, and mean the same thing:
$(\frac{20}{5})$$(\frac{x^{3}}{ x^{0}})$ $-$ $(\frac{30}{5})$$(\frac{x^{2}}{ x^{0}})$ $+$ $(\frac{5}{5})$$(\frac{x}{ x^{0}})$$+$ $(\frac{5}{5})$
OR
$(\frac{20}{5})$$(\frac{x^{3}}{1})$ $-$ $(\frac{30}{5})$$(\frac{x^{2}}{1})$ $+$ $(\frac{5}{5})$$(\frac{x}{1})$$+$ $(\frac{5}{5})$
• Next, we need to divide the coefficients and exponents for each term. However, "dividing" each term's exponents (and base) will leave us with the numerator every time in this problem, so now we only need to divide each term's coefficients:
$(\frac{20}{5})$ $=4$
$-$ $(\frac{30}{5})$ $=-6$
$(\frac{5}{5})$ $=1$
$(\frac{5}{5})$ $=1$
With each term's coefficient properly divided, we take those answers and throw in each term's respective "x" base and exponents:
Term #1: $x^{3}$
Term #2: $x^{2}$
Term #3: $x$
Term #4: none
Thus, giving us the answer of:
$4x^{3} - 6x^{2} + x + 1$