Question

How do you find a polynomial function that has zeros 0, -2, -3?

Answer 1

Hint: A function, in general, is an operator that gives an output when an input is given. A polynomial function is in the form $f\left( x \right)={{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+....+{{a}_{2}}{{x}^{2}}+{{a}_{1}}x+0$. Here x is the variable and a is the coefficient. A polynomial with 3 roots will be in the form $f\left( x \right)={{a}_{3}}{{x}^{3}}+{{a}_{2}}{{x}^{2}}+ax+c$. Here c is a constant.

Complete Step by Step Solution:
A zero of a polynomial is the value when substituted in the given polynomial function will give zero.
When the zeros 0, -2, and -3 are substituted for this polynomial function, it will give zero as an output. Therefore they can be written in terms of the following factors.
$\Rightarrow f\left( x \right)=\left( x \right)\left( x+2 \right)\left( x+3 \right)$
When we substitute $x=0$, the first factor will become zero and so the result will be zero. Similarly, if we apply -2 in the above function, the second factor will become zero and so the output will also be zero. The same applies to the third zero in which the third factor will become zero.
Therefore we now have a factored form of the required polynomial. If we multiply the factors together, we will get the polynomial function.
$\Rightarrow f\left( x \right)=\left( {{x}^{2}}+2x \right)\left( x+3 \right)=\left( {{x}^{3}}+2{{x}^{2}}+3{{x}^{2}}+6x \right)$

Hence the polynomial function that has zeros 0, -2, and -3 is
$\Rightarrow f\left( x \right)={{x}^{3}}+5{{x}^{2}}+6x$

Note:
While working with polynomial and quadratic equations, we should check whether the output is zero when the roots are substituted in the equation. Consider the polynomial equation $y={{x}^{3}}+5{{x}^{2}}+6x$ we just derived. If we apply the given roots 0, -2, and -3, it will give an output $y=0$. In this way, we can confirm that we have the right expression in our equation.