Question

Show that -1 is a zero of the polynomial $2{{x}^{3}}-{{x}^{2}}+x+4$.

Answer 1

Hint: Here, for checking -1 is a zero of the given polynomial or not, we may simply put -1 in place of x in the polynomial and check whether its value comes out to be zero or not. Since, it is given that we have to show that -1 is a zero of this polynomial, so its value must come zero.

Complete step-by-step answer:
Since, the given polynomial is:
$2{{x}^{3}}-{{x}^{2}}+x+4$
We know that by the term zero of a polynomial we mean that at that particular value, the value of the polynomial will be equal to zero.
It means if we put the value of zero of the polynomial in the place of the variable in the polynomial, we will get zero.
So, here we will apply this concept.
So, if we put x=-1 in the given polynomial its value will become zero because -1 is a zero of this polynomial.
So, on substituting x = -1 we get:
$\begin{align}
  & 2\times {{\left( -1 \right)}^{3}}-{{\left( -1 \right)}^{3}}+\left( -1 \right)+4 \\
 & =2\times \left( -1 \right)-1-1+4 \\
 & =-2-2+4 \\
 & =0 \\
\end{align}$
So, at x=-1 the value of this polynomial comes out to be zero. It proves that x = -1 is a zero of the given cubic polynomial.
Hence, we have shown that -1 is a zero of the given polynomial.

Note: Students should note here that another way of checking whether -1 is a zero of the given polynomial or not is that we may draw the graph of the polynomial. If the graph cuts the x-axis at x =-1 then -1 will be a zero of this polynomial.