To find a polynomial [tex]\( P(x) \)[/tex] of degree 10 with a leading coefficient of 1, and given roots with certain multiplicities, we must construct the polynomial based on these properties.
Here are the steps to form the polynomial:
1. Identify the Roots and Their Multiplicities:
- The root 0 has a multiplicity of 4.
- The root 1 has a multiplicity of 3.
- The root 3 has a multiplicity of 3.
2. Construct the Polynomial using the Roots and Multiplicities:
- For a root [tex]\( x = 0 \)[/tex] with a multiplicity of 4, the factor is [tex]\((x - 0)^4\)[/tex].
- For a root [tex]\( x = 1 \)[/tex] with a multiplicity of 3, the factor is [tex]\((x - 1)^3\)[/tex].
- For a root [tex]\( x = 3 \)[/tex] with a multiplicity of 3, the factor is [tex]\((x - 3)^3\)[/tex].
3. Combine the Factors:
- Multiply these factors together to construct the polynomial. Since the leading coefficient is 1, the polynomial can be written directly as the product of these factors.
Therefore, the polynomial [tex]\( P(x) \)[/tex] can be written as:
[tex]\[ P(x) = (x - 0)^4 \times (x - 1)^3 \times (x - 3)^3 \][/tex]
So, a possible formula for [tex]\( P(x) \)[/tex] is:
[tex]\[
P(x) = (x - 0)^4 (x - 1)^3 (x - 3)^3
\][/tex]
