To construct the polynomial \( p(x) \) with the specified zeros and multiplicities, we start with the zeros 1, 2, and 3 corresponding to the multiplicities 3, 2, and 1.
Part a:
The general form of the polynomial can be written as:
\[ p(x) = k(x - 1)^3(x - 2)^2(x - 3)^1 \]
where \( k \) is a constant that will be determined in part (b).
The degree of the polynomial \( p(x) \) can be found by adding the multiplicities of the zeros:
- The multiplicity of zero 1 is 3.
- The multiplicity of zero 2 is 2.
- The multiplicity of zero 3 is 1.
Thus, the degree of the polynomial is:
\[ \text{Degree} = 3 + 2 + 1 = 6 \]
So the degree is 6.
Part b:
Now, we want to find the specific polynomial \( p(x) \) such that \( p(0) = 6 \).
First, we substitute \( x = 0 \) into the polynomial:
\[ p(0) = k(0 - 1)^3(0 - 2)^2(0 - 3)^1 \]
Calculating this gives:
\[ p(0) = k(-1)^3(-2)^2(-3) = k(-1)(4)(-3) = k(12) \]
We are given that \( p(0) = 6 \), thus:
\[ k(12) = 6 \]
To find \( k \):
\[ k = \frac{6}{12} = \frac{1}{2} \]
Now we can write the specific polynomial \( p(x) \):
\[ p(x) = \frac{1}{2}(x - 1)^3(x - 2)^2(x - 3) \]
Therefore, the polynomial \( p(x) \) is:
\[ p(x) = \frac{1}{2}(x - 1)^3(x - 2)^2(x - 3) \]
And the degree is 6.