Question

1. a. Write the general polynomial p(x) whose only zeros are 1, 2 and 3, with multiplicity 3, 2 and 1 respectively. What is its degree? b. find p(x) described in part (a)if p(0) = 6.

Answer 1

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.