How do you simplify and divide $\dfrac{{{y}^{3}}+3{{y}^{2}}-5y-4}{y+4}$ ?

Question

Vedantu Content Team · Accepted Answer

Hint:We have been given a fractional expression which indicates that the 3-degree polynomial in the numerator has to be divided by the linear one-degree polynomial in the denominator. We shall divide these two functions by the algebraic long division method. Using the algebraic long division method, we will be able to obtain both the quotient as well as the remainder of the division being performed. Complete step by step solution:Given that  $\dfrac{{{y}^{3}}+3{{y}^{2}}-5y-4}{y+4}$.The easiest method to perform division of higher order polynomials, that is, polynomials of degree three or more is the algebraic long division method.This method has a special form of representation. The dividend, divisor and quotient are placed around the division box as follows:$divisor\overset{quotient}{\mathop{\left| \!{\overline {\,  dividend \,}} \right. }}\,$ Likewise, in this problem we have,$$y+4\overset{{{y}^{2}}}{\mathop{\left| \!{\overline {\,  \begin{align}  & {{y}^{3}}+3{{y}^{2}}-5y-4 \\  & {{y}^{3}}+4{{y}^{2}} \\ \end{align} \,}} \right. }}\,$$ Now we shall subtract these terms and get:$$\begin{align}  & y+4\overset{{{y}^{2}}-y}{\mathop{\begin{align}  & \left| \!{\overline {\,  \begin{align}  & {{y}^{3}}+3{{y}^{2}}-5y-4 \\  & {{y}^{3}}+4{{y}^{2}} \\ \end{align} \,}} \right.  \\  & \text{       }\overline{\begin{align}  & -{{y}^{2}}-5y-4 \\  & -{{y}^{2}}-4y \\ \end{align}} \\ \end{align}}}\, \\  & \text{             } \\ \end{align}$$ Here, we shall again subtract these terms and we get,$$\begin{align}  & y+4\overset{{{y}^{2}}-y-1}{\mathop{\begin{align}  & \left| \!{\overline {\,  \begin{align}  & {{y}^{3}}+3{{y}^{2}}-5y-4 \\  & {{y}^{3}}+4{{y}^{2}} \\ \end{align} \,}} \right.  \\  & \text{       }\overline{\begin{align}  & -{{y}^{2}}-5y-4 \\  & -{{y}^{2}}-4y \\ \end{align}} \\ \end{align}}}\, \\  & \text{                      }\overline{\begin{align}  & -y-4 \\  & -y-4 \\  & \text{  }\overline{0+0} \\ \end{align}} \\  & \text{             } \\ \end{align}$$Since we have obtained zero now, thus we can stop here as we have finally obtained our quotient.Therefore, the solution of division of  $\dfrac{{{y}^{3}}+3{{y}^{2}}-5y-4}{y+4}$ is $${{y}^{2}}-y-1$$. Note: The method to perform algebraic long division is by developing the quotient term-by-term. We find a term whose multiplication with the first of the divisors is equal to the first term of the dividend. Further we multiply all the terms of the divisor with the chosen term and place them below their respective degree term present in the dividend. This is one step. We keep simplifying and repeating this step until we obtain the remainder equal to zero or equal to a constant number. Another point to remember is that we have to subtract the terms coming in each step.