Question

Find a polynomial function of degree 7 with -2 as a zero of multiplicity​ 3, 0 as a zero of multiplicity3 ​, and 2 as a zero of multiplicity 1.

Answer 1

Answer:

The polynomial is $p(x) = ax^3(x+2)^3(x-2)$, in which a is the leading coefficient.

Step-by-step explanation:

Zeros of a function:

Given a polynomial f(x), this polynomial has roots $x_{1}, x_{2}, x_{n}$ such that it can be written as: $a(x - x_{1})*(x - x_{2})*...*(x-x_n)$, in which a is the leading coefficient.

-2 as a zero of multiplicity​ 3

This means that:

$p(x) = (x-(-2))^3 = (x+2)^3$

0 as a zero of multiplicity 3  

Then also:

$p(x) = (x+2)^3(x-0)^3 = x^3(x+2)^3$

2 as a zero of multiplicity 1.

Then:

$p(x) = x^3(x+2)^3(x-2)$

Adding the leading coefficient:

$p(x) = ax^3(x+2)^3(x-2)$

The polynomial is $p(x) = ax^3(x+2)^3(x-2)$, in which a is the leading coefficient.