To solve this question, let's break down each part of the polynomial [tex]\( P(x) = 5x^2(x-1)^3(x+9) \)[/tex].
1. Identifying the Degree of the Polynomial:
The degree of a polynomial is the highest exponent of [tex]\( x \)[/tex] when the polynomial is fully expanded.
- The term [tex]\( x^2 \)[/tex] contributes a degree of 2.
- The term [tex]\( (x-1)^3 \)[/tex] contributes a degree of 3.
- The term [tex]\( (x+9) \)[/tex] contributes a degree of 1.
Adding these degrees together:
[tex]\[
\text{Degree} = 2 + 3 + 1 = 6
\][/tex]
2. Identifying the Zeros of the Polynomial:
The zeros of the polynomial are the values of [tex]\( x \)[/tex] that make [tex]\( P(x) = 0 \)[/tex].
- [tex]\( x^2 \)[/tex] has a zero at [tex]\( x = 0 \)[/tex].
- [tex]\( (x-1)^3 \)[/tex] has a zero at [tex]\( x = 1 \)[/tex].
- [tex]\( (x+9) \)[/tex] has a zero at [tex]\( x = -9 \)[/tex].
Therefore, the zeros are [tex]\( 0 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( -9 \)[/tex].
3. Finding the Multiplicities of the Zeros:
The multiplicity of a zero is the exponent of the corresponding factor in the polynomial.
- The zero [tex]\( 0 \)[/tex] corresponds to [tex]\( x^2 \)[/tex], which has a multiplicity of 2.
- The zero [tex]\( 1 \)[/tex] corresponds to [tex]\( (x-1)^3 \)[/tex], which has a multiplicity of 3.
- The zero [tex]\( -9 \)[/tex] corresponds to [tex]\( (x+9) \)[/tex], which has a multiplicity of 1.
Summarizing all this information:
- The polynomial [tex]\( P(x) \)[/tex] has a degree of 6.
- The zeros of the polynomial are [tex]\( 0 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( -9 \)[/tex].
- The zero [tex]\( 0 \)[/tex] has a multiplicity of 2.
- The zero [tex]\( 1 \)[/tex] has a multiplicity of 3.
Thus, the answers are as follows:
The polynomial [tex]\( P(x)=5 x^2(x-1)^3(x+9) \)[/tex] has degree [tex]\( 6 \)[/tex]. It has zeros [tex]\( 0, 1, \)[/tex] and [tex]\( -9 \)[/tex]. The zero [tex]\( 0 \)[/tex] has multiplicity [tex]\( 2 \)[/tex], and the zero [tex]\( 1 \)[/tex] has multiplicity [tex]\( 3 \)[/tex].
