give the quartic polynomial function with the roots: -3, -2, and 4 of multiplicity 2.

A quartic polynomial function with roots \(-3\), \(-2\), and \(4\) (with multiplicity 2 for the root at \(4\)) can be written as:

\[
f(x) = a(x + 3)(x + 2)(x - 4)^2
\]

Here’s the step-by-step explanation:

1. The root \(-3\) corresponds to the factor \((x + 3)\).
2. The root \(-2\) corresponds to the factor \((x + 2)\).
3. The root \(4\) with multiplicity 2 corresponds to the factor \((x - 4)^2\).

If you expand this expression:

\[
f(x) = a(x + 3)(x + 2)(x - 4)^2
\]

The constant \(a\) is a leading coefficient that can be any nonzero constant. For simplicity, we often set \(a = 1\), which gives the polynomial:

\[
f(x) = (x + 3)(x + 2)(x - 4)^2
\]

Now, expanding the polynomial:

1. Expand \((x - 4)^2\):

\[
(x - 4)^2 = x^2 - 8x + 16
\]

2. Now multiply \((x + 2)\) by \((x^2 - 8x + 16)\):

\[
(x + 2)(x^2 - 8x + 16) = x^3 - 8x^2 + 16x + 2x^2 - 16x + 32 = x^3 - 6x^2 + 32
\]

3. Multiply \((x + 3)\) by the result:

\[
(x + 3)(x^3 - 6x^2 + 32) = x^4 - 6x^3 + 32x + 3x^3 - 18x^2 + 96
\]

4. Combine like terms:

\[
f(x) = x^4 - 3x^3 - 18x^2 + 32x + 96
\]

So, the quartic polynomial is:

\[
f(x) = x^4 - 3x^3 - 18x^2 + 32x + 96
\]

Let me know if you'd like further details or have any questions.

Here are 5 related questions:
1. How do you determine the multiplicity of a root?
2. How do leading coefficients affect polynomial graphs?
3. Can a polynomial have more than one root with multiplicity?
4. What is the relationship between the degree of a polynomial and the number of roots?
5. How do you factor polynomials with higher-degree terms?

**Tip:** When expanding polynomials, handle the terms step by step to avoid mistakes, especially with multiplicity roots.

Discover how to find the quartic polynomial with roots -3, -2, and 4 (with multiplicity 2 for the root at 4). This step-by-step guide explores polynomial functions, multiplicity, and root expansion. Suitable for students in Grades 10-12.

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