Question

Find a polynomial function with leading coefficient 1 that has the given zeros, multiplicities, and degree. Zero: 2, multiplicity: 3 Zero: 0, multiplicity: 2 Degree: 5

Answer 1

Given:

Zero: 2, multiplicity: 3

Zero: 0, multiplicity: 2

Degree: 5

Leading coefficient = 1

To find:

The polynomial function.

Solution:

The general form of a polynomial is

$P(x)=a(x-c_1)^{m_1}(x-c_2)^{m_2}...(x-c_n)^{m_n}$

where, a is a constant, $c_1,c_2,...,c_n$ are zeroes with multiplicity $m_1,m_2,...,m_n$.

Using the given information and the general form of a polynomial, we get

$P(x)=a(x-2)^{3}(x-0)^{2}$

$P(x)=ax^2(x^3-6x^2+12x-8)$

$P(x)=a(x^5-6x^4+12x^3-8x^2)$

Leading coefficient is 1, so the value of a is also 1.

$P(x)=1(x^5-6x^4+12x^3-8x^2)$

$P(x)=x^5-6x^4+12x^3-8x^2$

Therefore, the required polynomial is $P(x)=x^5-6x^4+12x^3-8x^2$.