91Ó°ÊÓ

The degree of a polynomial is the highest power of the variable in the polynomial expression. In the given exercise, we are dealing with a polynomial of degree 4. This means the highest exponent of our variable, usually denoted as 'x', will be 4. The degree of a polynomial helps to determine its general shape and the number of roots (or zeros) it can have.
For example, a degree 4 polynomial can have up to 4 zeros, although some zeros may have multiplicity greater than 1. Understanding the degree provides insight into the complexity and behavior of the polynomial.

The multiplicity of a zero refers to the number of times a particular zero appears in the polynomial. In other words, it indicates how many times the corresponding factor is repeated in the factored form of the polynomial.
In our problem, the polynomial has zeros at 1 and \( \frac{3}{2} \), with each having a multiplicity of 1, and a zero at 0 with multiplicity 2. This tells us that:

• The factor \( (x-1) \) appears once.
• The factor \( \left(x - \frac{3}{2} \right) \) appears once.
• The factor \( x \) appears twice, giving us \( x^2 \).

Understanding multiplicity is important because it affects the shape of the polynomial's graph near its zeros.

The factored form of a polynomial expresses it as a product of its factors. This form is particularly useful for identifying zeros and their multiplicities. The given polynomial, with the zeros and multiplicities specified, can be written in factored form as:
\[ f(x) = C \times x^2 \times (x-1) \times \left(x-\frac{3}{2} \right) \] Here, we use C as a constant coefficient, which can be adjusted based on additional conditions.
The factored form makes it easier to see the polynomial's roots directly, and it provides a foundation for further operations like expanding to find the standard form.

Expanding a polynomial involves multiplying out the factored form to express the polynomial as a standard sum of terms. In our case, we start with:
\[ f(x) = C \times x^2 \times (x-1) \times \left(x-\frac{3}{2} \right) \]
First, we expand the factors \( (x-1) \) and \( \left(x - \frac{3}{2} \right) \):
\[ (x-1)(x-\frac{3}{2}) = x^2 - x - \frac{3}{2}x + \frac{3}{2} = x^2 - \frac{5}{2}x + \frac{3}{2} \]
Next, multiply this by \( x^2 \):
\[ f(x) = C \times x^2 \times \left( x^2 - \frac{5}{2}x + \frac{3}{2} \right) = C \left( x^4 - \frac{5}{2}x^3 + \frac{3}{2}x^2 \right) \]
Finally, assuming \( C = 1 \), the expanded polynomial is:
\[ f(x) = x^4 - \frac{5}{2}x^3 + \frac{3}{2}x^2 \]
Expanding helps to transform the polynomial into a more usable form for graphing and further analysis.