Question

Construct a polynomial function with the following properties: third degree, 1 is a zero of multiplicity 1, −4 is the only other zero, leading coefficient is 5.

Answer 1

We want to construct a polynomial for the given information.

The polynomial is:

p(x) = 5*(x - 1)*(x + 4)^2

Let's see how to get that:

We know that a polynomial of leading coefficient A, with zeros {x₁, x₂, ..., xₙ} each with multiplicity {m₁, ..., mₙ}

Can be written as:

$p(x) = A*(x - x_1)^{m_1}*...*(x - x_n)^{m_n}$

Here we know:

Leading coefficient = 5

One of the zeros is 1, with multiplicity 1.

The other zero is -4, and we don't know the multiplicity.

But we know that the degree of the polynomial is 3, and the sum of the multiplicities must be equal to the degree of the polynomial, then the multiplicity of this zero must be 2.

Now we can completely determine the polynomial, it is written as:

p(x) = 5*(x - 1)*(x + 4)^2

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