Question

Find each quotient.

$\left(5x^2{y}^2-10x^{2}y+5xy\right)÷5xy$

Short answer

The quotient is $xy-2x+1$.

Step-by-step solution

Step 1. Define the concept.

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Step 2. Divide each term by 5xy. 

First write as a fraction, then divide each term by $5xy$.

role="math" localid="1648065397746" $$\begin{gathered} \left(5x^2{y}^2-10x^{2}y+5xy\right)÷5xy=\frac{5x^2{y}^2-10x^{2}y+5xy}{5xy} \\ =\frac{5x^2{y}^2}{5xy}+\frac{\left(-10x^{2}y\right)}{5xy}+\frac{5xy}{5xy} \end{gathered}$$

Step 3. Simplify the expression.

Divide out the common factors and simplify.

$\begin{array}{c}\left(5x^2{y}^2-10x^{2}y+5xy\right)÷5xy=\frac{5x^2{y}^2}{5xy}+\frac{\left(-10x^{2}y\right)}{5xy}+\frac{5xy}{5xy}\\ =xy-\frac{10x^{2}y}{5xy}+1\\ =xy-2x+1\end{array}$

Therefore, $\left(x^3+7x^2+10x-6\right)÷\left(x+3\right)$is $xy-2x+1$.