Question

A polynomial function h(x) has a zero of x = 3 - 41 with a multiplicity of one. Certain values of h() are given in the following table.
x h(x)
-5 0
-2 3
-1 0
1. 2
4 0
7 6
10 O
If every real x-intercept of h() is shown in the table and each has a multiplicity of one, what is the degree of h(x)?
3
4
5
6

Answer 1

The degree of a polynomial is the value of the highest degree of a monomial in the polynomial

The degree of the polynomial h(x) is 5

The reason the above value is correct is as follows;

Given;

A zero of a polynomial function h(x) is the imaginary number, x = 3 - 4·i

The multiplicity of the root at x = 3 - 4·i is One

The given table showing the real roots of the polynomial is presented as follows;

$\begin{array}{|c|cc|}\mathbf{x}&&\mathbf{h(x)}\\\displaystyle -5&&0\\-2&&3\\-1&&0\\1&&2\\4&&0\\7&&6\\10&&0\end{array}\right]$

Required:

To find the degree of the polynomial, h(x)

Solution:

The maximum number of roots of a polynomial is given by the degree of the polynomial

The number of roots is given by the number zeros

The maximum number of roots an nth degree polynomial can have = n roots

From the given table, the number of zeros = 4 (each with multiplicity of one) = The number of real roots

Therefore, the total number of roots of the polynomial = 4 + 1 = 5 = The (minimum possible) degree of the polynomial

The degree of the polynomial = 5

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