Algebra: A Combined Approach (4th Edition)

$5x + 1$
Given $(25x^{3} + 5x^{2}) \div 5x^{2}$, the answer, simplified, is $5x + 1$. • The 1st step in dividing the polynomial by the monomial is to set up the equation in a more visible format: $\frac{(25x^{3} + 5x^{2})}{5x^{2}}$ • Next, 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: $25x^{3}$ Term #2: $5x^{2}$ • Then we need to set up each of those terms as the numerator, with the monomial as the divisor (the denominator, or the bottom half of the equation above), which is $5x^{2}$. Doing so now makes the equation look like this: $\frac{25x^{3}}{5x^{2}}$ + $\frac{5x^{2}}{5x^{2}}$ • Now we need to simplify each fraction, consisting of the one term of the polynomial and the monomial. Let’s start with the first one, and let’s name it Term “A” to make it easier to see how each term is simplified. Term “A”: $\frac{25x^{3}}{5x^{2}}$ STEP #1: A). We first need to divide the coefficients given (the numbers in front of x), then we need to divide the exponents. So, let’s set it up like this to make it a little easier: $(\frac{25}{5})$ $(\frac{x^{3}}{ x^{2}})$ STEP #2: A.) So, let’s divide the coefficients first, and then the exponents: $(\frac{25}{5})$ $25\div5 = 5$ STEP #3: A.) Now, let’s divide the variable's exponents to further simplify our fraction: $(\frac{x^{3}}{ x^{2}})$ Remember that when we are "dividing" exponents, we are really subtracting the denominator's exponent (which is 2) from the numerator's exponent (which is 3). So, $x^{3-2}$ $=x^{1}$, simplified to $x$ STEP #4: A.) Combine the answers for the simplified fraction of term "A" as shown in steps #2 and #3: Term “A” $\frac{25x^{3}}{5x^{2}}$, which we rewrote as: $(\frac{25}{5})$ $(\frac{x^{3}}{ x^{2}})$ is now simplified to: $5x$ • Now, we move on to simplifying the polynomial’s 2nd term, which we set up above: $\frac{5x^{2}}{5x^{2}}$ Let’s name it Term “B” to make it easier to see how the term is simplified. STEP #1: B.) We first need to divide the coefficients given (the numbers in front of x), then we need to divide the exponents. So, let’s set it up like this to make it a little easier: $(\frac{5}{5})$$(\frac{x^{2}}{x^{2}}) STEP #2: B.) So, let’s divide the coefficients first, and then the exponents: (\frac{5}{5}) 5\div5 = 1 STEP #3: B.) Now, let’s divide the variable's exponents to further simplify our fraction: (\frac{x^{2}}{x^{2}}) Remember that when we are "dividing" exponents, we are really subtracting the denominator's exponent (which is 2) from the numerator's exponent (which is 2). So, x^{2-2} =x^{0}, simplified to 1, because anything to the 0 power is 1. STEP #4: A.) Combine the answers for the simplified fraction of term "B" as shown in steps #2 and #3: Term “B” \frac{5x^{2}}{5x^{2}}, which we rewrote as (\frac{5}{5})$$(\frac{x^{2}}{x^{2}})$ is now simplified to: $(1)(1)$, which can be further simplified to $1$. • Now that we have simplified each polynomial's terms when each is divided by the monomial, the original equation of: $(25x^{3} + 5x^{2}) \div 5x^{2}$, is now $5x + 1$. So, the answer is $5x + 1$.