Question

Find the product of zero of cubic polynomials $p(x)={{x}^{3}}+4{{x}^{2}}+x-6$.

Answer 1

Hint: We know the general form of cubic polynomial which is $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$. So in this $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$ we have $\alpha ,\beta ,\delta $ as cube root of $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$. We know that the product of these roots is equal to $\dfrac{-d}{a}$.

Complete step-by-step answer:
We are given the cubic polynomial $p(x)={{x}^{3}}+4{{x}^{2}}+x-6$.
We have to find the product of zeros of cubic polynomials.
So now first let us compare $p(x)={{x}^{3}}+4{{x}^{2}}+x-6$ to $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$.
We get, $a=1,b=4,c=1,d=-6$.
We know that, $\alpha ,\beta ,\delta $ are roots of $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$.
So, the product of zeros of polynomials$=\dfrac{-d}{a}$.
For $p(x)={{x}^{3}}+4{{x}^{2}}+x-6$, the product of zeros of polynomial$=\dfrac{-(-6)}{1}$.
So, the product of zeros of polynomial$=6$.
Therefore, the product of zeros of polynomials ${{x}^{3}}+4{{x}^{2}}+x-6$ is $6$.

Additional information:
Roots of polynomials are the solutions for any given polynomial for which we need to find the value of the unknown variable. If we know the roots, we can evaluate the value of the polynomial to zero. Each variable separated with an addition or subtraction symbol in the expression is better known as the term. The degree of the polynomial is defined as the maximum power of the variable of a polynomial. A polynomial with only one term is known as a monomial. A monomial containing only a constant term is said to be a polynomial of zero degrees. A polynomial can account to null value even if the values of the constants are greater than zero. All cubic equations have either one real root, or three real roots. If the polynomials have the degree three, they are known as cubic polynomials.

Note: The cubic polynomial is a polynomial with the highest degree of 3. The cubic polynomial should be in the form of $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$, where $a\ne 0$. Let say $\alpha ,\beta ,\delta $are the three zeros of a polynomial, then
Sum of zeros, $\alpha +\beta +\delta =\dfrac{-b}{a}$,
Product of zeros, $\alpha \beta \delta =\dfrac{-d}{a}$.