The polynomial 𝑃(𝑥)  of degree 4 has

a root of multiplicity 2 at x = 3

a root of multiplicity 1 at x = 0 and at x = -2

It goes through the point (5,42)

Find a formula for 𝑃(𝑥)=


To find the polynomial \( P(x) \) of degree 4 with the given properties, we will use the following information:

1. **Degree 4**: The polynomial is of degree 4.
2. **Root of multiplicity 2 at \( x = 3 \)**: The factor \( (x - 3) \) will appear squared.
3. **Root of multiplicity 1 at \( x = 0 \)**: The factor \( x \) will appear once.
4. **Root of multiplicity 1 at \( x = -2 \)**: The factor \( (x + 2) \) will appear once.
5. **The polynomial goes through the point \( (5, 42) \)**: This means \( P(5) = 42 \).

Given this, the polynomial can be expressed as:

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

where \( a \) is a constant to be determined.

### Step 1: Expand the Factors

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

First, expand \( (x - 3)^2 \):

\[
(x - 3)^2 = (x - 3)(x - 3) = x^2 - 6x + 9
\]

Then, multiply by \( x \):

\[
x(x^2 - 6x + 9) = x^3 - 6x^2 + 9x
\]

Finally, multiply by \( (x + 2) \):

\[
P(x) = a(x^3 - 6x^2 + 9x)(x + 2)
\]

\[
= a(x^4 - 6x^3 + 9x^2 + 2x^3 - 12x^2 + 18x)
\]

\[
= a(x^4 - 4x^3 - 3x^2 + 18x)
\]

\[
= a(x^4 - 4x^3 - 3x^2 + 18x)
\]

### Step 2: Determine \( a \) Using the Point \( (5, 42) \)

To find \( a \), we use the point \( (5, 42) \), which tells us that \( P(5) = 42 \). Substitute \( x = 5 \) and set \( P(5) = 42 \):

\[
P(5) = a(5^4 - 4(5)^3 - 3(5)^2 + 18(5))
\]

\[
42 = a(625 - 500 - 75 + 90)
\]

\[
42 = a(140)
\]

\[
a = \frac{42}{140} = \frac{3}{10} = 0.3
\]

### Final Polynomial

\[
P(x) = 0.3(x^4 - 4x^3 - 3x^2 + 18x)
\]

\[
P(x) = 0.3x^4 - 1.2x^3 - 0.9x^2 + 5.4x
\]

Thus, the polynomial is:

\[
\boxed{P(x) = 0.3x^4 - 1.2x^3 - 0.9x^2 + 5.4x}
\]

Would you like a more detailed explanation or have any questions?

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### Questions

1. What does it mean for a root to have a multiplicity greater than 1?
2. How can we verify that the polynomial goes through a given point?
3. Why do we need to find the constant \( a \) in the polynomial?
4. How does the multiplicity of a root affect the graph of the polynomial at that root?
5. What are the general steps for expanding a polynomial with multiple factors?
6. How can you find the y-intercept of a polynomial?
7. How does the degree of a polynomial affect its end behavior?
8. What role does the leading coefficient play in the shape of a polynomial's graph?

### Tip

Always ensure that the number of roots (counting multiplicities) matches the degree of the polynomial to confirm correctness.

Learn how to find the polynomial P(x) of degree 4 with roots of specified multiplicities at x = 3, x = 0, and x = -2, and passing through the point (5, 42). Detailed step-by-step solution provided.

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