Questions: The polynomial of degree 5, P(x) has leading coefficient 1, has roots of multiplicity 2 at x=3 and x=0, and a root of multiplicity 1 at x=-5. Find a possible formula for P(x).

The polynomial of degree 5, P(x) has leading coefficient 1, has roots of multiplicity 2 at x=3 and x=0, and a root of multiplicity 1 at x=-5. Find a possible formula for P(x).

Transcript text: The polynomial of degree $5, P(x)$ has leading coefficient 1, has roots of multiplicity 2 at $x=3$ and $x=0$, and a root of multiplicity 1 at $x=-5$. Find a possible formula for $P(x)$.

Solution

Solution Steps

Step 1: Identify the Roots and Their Multiplicities

Root: 3 with multiplicity 2
Root: 0 with multiplicity 2
Root: -5 with multiplicity 1

Step 2: Construct the Polynomial

The polynomial is constructed as follows based on the roots and their multiplicities: \[P(x) = 1 * (x - (3))^2 * (x )^2 * (x - (-5))^1\]

Step 3: Adjust for the Leading Coefficient

The leading coefficient is already considered in the construction of the polynomial. Thus, the leading coefficient is 1.

Final Answer:

The polynomial of degree 5 with leading coefficient 1, having roots at specified locations with given multiplicities is: \[P(x) = 1 * (x - (3))^2 * (x )^2 * (x - (-5))^1\] Note: For a simplified form, further expansion and simplification of the polynomial might be required, which is not covered in this solution.

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