Problem 2 - College Algebra with Corequisite Support

Question

If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

Step-by-Step Solution

Answer

Identify and add the multiplicities of all factors.

Step 1: Identify Each Factor

Look at the polynomial in its factored form and identify each individual factor. A factor in this context is a term that can be multiplied by other terms to achieve the polynomial expression.

Step 2: Count the Multiplicity of Each Factor

For each factor found in the polynomial, determine the multiplicity, which is the exponent of the factor. If a factor appears as \((x-a)^n\), then the multiplicity of this factor is \(n\).

Step 3: Sum of Multiplicities

Add up all the multiplicities of the factors you identified. The sum of the multiplicities will give you the degree of the polynomial function.

Key Concepts

Factored Form

When dealing with polynomials, the factored form is a way of expressing the polynomial as a product of its factors. This form provides significant insights into the behavior and characteristics of the polynomial function.

• Factored form breaks down a polynomial into simpler multiplicative components.
• Factors in a polynomial are expressions that multiply together to form the entire polynomial.
• This form is particularly useful because it makes certain properties of the polynomial, such as roots and multiplicities, readily apparent.

A polynomial in factored form might look like this: \( (x-a)(x-b)^2(x-c)^3 \). Each group, such as \((x-a)\), is a factor of the polynomial. The factored form is not only helpful in identifying roots or zeros of the polynomial but also in determining other features like the leading coefficient. It streamlines the process of analyzing and solving polynomial equations.

Degree of Polynomial

The degree of a polynomial is a crucial component in understanding its overall form and behavior. It is defined as the highest power of the variable present in the polynomial expression.

• The degree determines how many solutions or "roots" a polynomial can have.
• It also hints at the end behavior of the polynomial function; for example, whether it will go to infinity or negative infinity as the input grows large.

To find the degree of a polynomial in factored form, follow these simple steps:1. Identify the factors: As explained earlier, look at the polynomial written in factored form and list all factors. 2. Determine the multiplicity of each factor: This is the exponent applied to each factor.3. Add the multiplicities: The sum of these multiplicities gives you the degree of the polynomial.As an example, if we have \( (x-1)^2(x+3) \), the degree would be \( 2 + 1 = 3 \). The degree tells us that the graph will curve up and down a maximum of two times (since its maximum number of turning points is always one less than its degree). Understanding a polynomial's degree provides significant insight into its potential complexity.

Multiplicity

Multiplicity is another important concept tied closely to polynomial functions expressed in factored form.

• Multiplicity refers to how many times a particular root is repeated in the polynomial's factorization.
• It is indicated by the exponent of a factor in the polynomial.

For instance, in \( (x+2)^3 \), the root \( x = -2 \) has a multiplicity of \( 3 \). This means the root is repeated three times. Understanding multiplicity is important because:

• If the multiplicity of a root is odd, the graph of the polynomial will cross the x-axis at this root.
• If it is even, the graph will only touch the x-axis and turn back.

Multiplicity affects not only the shape of the graph around the roots but also how the polynomial behaves as x moves towards positive or negative infinity.