Question

How do you find a polynomial function of degree $4$ with $ - 4$ as a zero of multiplicity $3$ and $0$ as a zero of multiplicity $1$?

Answer 1

Hint: In order to solve this, first we write out all the given variables. According to our question, our polynomial should have degree of $4$ and should have factors where $x = - 4$ and $x = 0$. $x = - 4$has a multiplicity of $3$, therefore we raise it to the power of $3$ and $x = 0$ has a multiplicity of $1$ so we raise it to $1$. After getting our required factors, we solve it further to get our polynomial in the standard form.

Complete step-by-step solution:
We need to find a polynomial function of degree $4$.
We first find the factors of the required polynomial. Here in the given, we have:
$x = - 4$ , with a multiplicity of $3$ …….....(A)
$x = 0$ , with a multiplicity of $1$ …………..(B)
Now in order to make these into factors, we bring the numerical terms to the right hand side:
$x + 4 = 0$
Since, (B) already has $0$ on the right hand side; therefore we need not do anything further
Our polynomial is represented by $f\left( x \right)$.
Now, we simply multiply our given factors, with each factor raised to the power of the multiplicity given in the question for that particular factor.
Factor $\left( {x + 4} \right)$ is raised to the power $3$ and $x$ is raised to the power $1$ :
$f\left( x \right) = {\left( {x + 4} \right)^3}\left( x \right)$ ………….equation (C )
Now, we solve this further by expanding the formulas.
We know that ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$
Thus ${\left( {x + 4} \right)^3} = {x^3} + {\left( 4 \right)^3} + \left[ {3 \times \left( x \right) \times \left( 4 \right)} \right]\left( {x + 4} \right)$
$ \Rightarrow {\left( {x + 4} \right)^3} = {x^3} + 64 + 12x\left( {x + 4} \right)$
We solve this further:
$ \Rightarrow {\left( {x + 4} \right)^3} = {x^3} + 64 + 12{x^2} + 48x$
We put the value of ${\left( {x + 4} \right)^3}$ in equation (C), we get:
$ \Rightarrow f\left( x \right) = \left( {{x^3} + 64 + 12{x^2} + 48x} \right)\left( x \right)$
We solve it further:
$ \Rightarrow f\left( x \right) = {x^4} + 64x + 12{x^3} + 48{x^2}$
Let’s arrange the above polynomial according to the descending powers:
 $ \Rightarrow f\left( x \right) = {x^4} + 12{x^3} + 48{x^2} + 64x$
Thus, we have our required polynomial of degree $4$ in the standardized form.

Note: A polynomial is mathematical expression that is composed of numerical, variables, etc joined with mathematical operations such as addition, subtraction, multiplication and division. There are four types of polynomials depending on the number of variables present in it such as:
Monomials (With One Term)
Binomial (With Two Terms In It)
Trinomial (Three Terms In One Expression)
Polynomial Has More Than Three Terms.