Question

A polynomial function has a root of -4 with multiplicity 4, a root of -1 with multiplicity 3, and a root of 5 with multiplicity 6. If
the function has a positive leading coefficient and is of odd degree, which could be the graph of the function?
a
8
6
4
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Answer 1

Using the Factor Theorem, the polynomial is given by: $f(x) = (x + 4)^4(x + 1)^3(x - 6)^6$

• The graph is sketched at the end of the answer.

The Factor Theorem states that a polynomial function with roots $x_1, x_2, \cdots, x_n$ is given by:

$f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)$

In which a is the leading coefficient.

In this problem, the roots are:

• Root of -4 with multiplicity 4, hence $x_1 = x_2 = x_3 + x_4 = -4$.

• Root of -1 with multiplicity 3, hence $x_5 = x_6 = x_7 = 3$.

• Root of 5 with multiplicity 6, hence $x_8 = x_9 = x_10 = x_11 = x_12 = x_13 = 6$

Then:

$f(x) = a(x - (-4))^4(x - (-1))^3(x-6)^6$

$f(x) = a(x + 4)^4(x + 1)^3(x - 6)^6$

• Positive leading coefficient, hence $a = 1$.

• 13th degree, so it is odd.

Then:

$f(x) = (x + 4)^4(x + 1)^3(x - 6)^6$

At the end of the answer, an sketch of the graph is given.

For more on the Factor Theorem, you can check brainly.com/question/24380382