Question

A polynomial function has a zero at 4 (multiplicity 3) and 0 (multiplicity 1). write a function in standard form

Answer 1

Y=3x(x-4)
Hope this helps!!

Answer 2

Answer:

The function is :

$f(x)=x^{4}-12x^{3}+48x^{2}-64x$

Step-by-step explanation:

For a polynomial, if $x=a$ is a zero of the function, then $(x-a)$ is a factor of the function.

Also if $x=a$ is a zero of the function with multiplicity $k$ , then $(x-a)^{k}$ is a factor of the function.

We can write the function as the product of the factors using its zeros.

$x=4$ is a zero with multiplicity 3 ⇒ $(x-4)^{3}$ is a factor of the function

$x=0$ is a zero with multiplicity 1 ⇒ $(x-0)^{1}=x^{1}=x$ is a factor of the function

Finally, we can write the function $f(x)$ as the product of its factors :

$f(x)=x.(x-4)^{3}$

If we work with the expression of $f(x)$ :

$x.(x-4)^{3}=x.[x^{3}+3x^{2}(-4)+3x(-4)^{2}+(-4)^{3}]$

⇒

$x.(x^{3}-12x^{2}+48x-64)$ = $x^{4}-12x^{3}+48x^{2}-64x$

The function written in standard form is :

$f(x)=x^{4}-12x^{3}+48x^{2}-64x$