Question

Using the factor theorem, factorize each of the following polynomials: ${{y}^{3}}-7y+6$ ?

Answer 1

Hint: In the given question we need to find the factors of the polynomial of three degrees. So, for that first of all we will consider the constant term of the polynomial and then find the factors of the constant term and then identify the roots of the polynomial and hence find the factors.

Complete step-by-step answer:
In the given question we have a three-degree polynomial which is $f\left( y \right)={{y}^{3}}-7y+6$ and here we can clearly see that the constant term is 6. Now, we know that the factors of +6 are $\pm 1,\pm 2,\pm 3,\pm 6$.
Now, we need to find any three roots of the polynomial which would be from these factors therefore, if $y=1$ then ${{1}^{3}}-7\times 1+6=0$ .This implies that $y=1$ is the solution of the polynomial therefore (y-1) is the factor. Now, let us see for y=2 then ${{2}^{3}}-7\times 2+6\Rightarrow 8+6-14=0$, this implies (y-2) is also the factor of the polynomial and now let us check for y=-3 then ${{\left( -3 \right)}^{3}}-7\times \left( -3 \right)+6\Rightarrow -27+6+21=0$this implies that (y+3) is also the factor of the given polynomial. Now we know that since the degree of the polynomial is 3 therefore it can have maximum three linear factors.
Therefore,
 $f\left( y \right)=k\left( y-1 \right)\left( y-2 \right)\left( y+3 \right)$
${{y}^{3}}-7y+6=k\left( y-1 \right)\left( y-2 \right)\left( y+3 \right)$
Now putting y=0 we have
$\begin{align}
  & 6=6k \\
 & \Rightarrow k=1 \\
\end{align}$
Therefore, $f\left( y \right)=k\left( y-1 \right)\left( y-2 \right)\left( y+3 \right)$implies $f\left( y \right)=\left( y-1 \right)\left( y-2 \right)\left( y+3 \right)$.
Hence, ${{y}^{3}}-7y+6=\left( y-1 \right)\left( y-2 \right)\left( y+3 \right)$.

Note: We need to be careful while finding the roots and also remember that the number of factors is equal to the degree of the polynomial which makes it easy for us to identify the number of solutions. Also, we must be careful while finding the constant value.