Question

If $y-3$ is a factor of ${{y}^{3}}+2{{y}^{2}}-9y-18$, then find the other two factors.

Answer 1

Hint: Here one factor is given, we can find the other factors by polynomial division. First reduce the 3rd degree polynomial into a 2nd degree polynomial and then by reducing the 2nd degree polynomial we can find the other two factors.

Complete step-by-step solution -

Here, it is given that $y-3$ is a factor of the polynomial ${{y}^{3}}+2{{y}^{2}}-9y-18$. We have to find the other two factors.
First, we have to reduce the 3rd degree polynomial into a 2nd degree polynomial by polynomial division. The polynomial division is given by:
$y-3\overset{{{y}^{2}}+5y+6}{\overline{\left){\begin{align}
  & {{y}^{3}}+2{{y}^{2}}-9y-18 \\
 & {{y}^{3}}-3{{y}^{2}} \\
 & \overline{0{{y}^{3}}\text{ +}5{{y}^{2}}-9y} \\
 & \text{ }5{{y}^{2}}-15y \\
 & \text{ }\overline{0{{y}^{2}}\text{ }+6y-18} \\
 & \text{ }6y-18 \\
 & \text{ }\overline{0y-0}\text{ } \\
\end{align}}\right.}}$
Therefore, by dividing the polynomial ${{y}^{3}}+2{{y}^{2}}-9y-18$ by $y-3$ we got the quotient as ${{y}^{2}}+5y+6$. i.e the 3rd degree polynomial is now reduced into a second degree polynomial .
Next, we can write the polynomial as:
${{y}^{3}}+2{{y}^{2}}-9y-18=(y-3)\left( {{y}^{2}}+5y+6 \right)$ ….. (1)
Now, we have to find the other two factors from the polynomial ${{y}^{2}}+5y+6$.
Consider ${{y}^{2}}+5y+6$ split $5y$as $2y+3y$. i.e we can write:
${{y}^{2}}+5y+6={{y}^{2}}+2y+3y+6$
For the above equation $y$ is common for the first two terms, so take $y$ outside and 3 is common for the last two terms, take 3 outside.
Hence, our equation becomes:
${{y}^{2}}+5y+6=y(y+2)+3(y+2)$
From the above equation since, $y+2$ is common, take it outside, we obtain:
${{y}^{2}}+5y+6=(y+2)(y+3)$
In the next step, we have to substitute the factors of the polynomial ${{y}^{2}}+5y+6$, $y+2$ and $y+3$ in equation (1). i.e. our equation (1) becomes:
${{y}^{3}}+2{{y}^{2}}-9y-18=(y-3)(y+2)(y+3)$
Therefore, the three factors of the polynomial ${{y}^{3}}+2{{y}^{2}}-9y-18$ are $y-3,y+2$ and $y+3$.

Note: Here in place of polynomial division you can also apply synthetic division to reduce the polynomial. If you know how to do synthetic division it is much simpler than polynomial division.