Problem 80 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polynomial function with the given zeros, multiplicities, and degree. (There are many correct answers.) Zero: \(1,\) multiplicity: 2 Zero: \(4,\) multiplicity: 2 Degree: Falls to the left, Falls to the right

Step-by-Step Solution

Answer

The polynomial that meets these conditions is \(x^4 - 10x^3 + 35x^2 - 50x + 16\).

Step 1 - Find the Basic Polynomial

Knowing the zeros and their multiplicities, a basic polynomial can be formed. This is done by taking (x - zero) to the power of multiplicity for each zero. So for this problem, with zeros 1 and 4 with multiplicities of 2, this gives us \((x - 1)^2 * (x - 4)^2\).

Step 2 - Apply Degree Rules

The degree of a polynomial affects its end behavior. If the degree is odd, the end behavior is the opposite on the left and right. In this case, the polynomial falls to the left and falls to the right, which indicates the degree should be even. Since the degree from step 1 is 4, which is even, no adjustments are needed. Thus, the polynomial is still \((x - 1)^2 * (x - 4)^2\).

Step 3 - Expand the Polynomial

Expand the polynomial to get a form that is easier to work with. This will give us \(x^4 - 10x^3 + 35x^2 - 50x + 16\).

Key Concepts

Zeros of a Polynomial

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, these are the solutions to the polynomial equation. For example, if you have a polynomial function \( f(x) \), the zeros are the values \( x \) where \( f(x) = 0 \). Finding the zeros is essential as they provide valuable information about the graph of the function.

In practice, to determine the zeros from a polynomial, one typically factors the expression or uses the quadratic formula for quadratic polynomials. Zerose are also roots of the polynomial and can be real or complex numbers. For our example, with given zeros of 1 and 4, when plugged into the polynomial equation, they result in zero, confirming that they are indeed zeros of the polynomial.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears in a polynomial equation. If a zero occurs more than once, we say it has a multiplicity greater than one. This is reflected in the factored form of the polynomial, where the term associated with that zero is raised to the power of its multiplicity.

In the given example, each zero, 1 and 4, has a multiplicity of 2. This is represented in the polynomial by \((x - 1)^2\) and \((x - 4)^2\). The multiplicity of a zero affects the shape of the graph at the root. For instance:

• If the multiplicity is odd, the graph crosses the x-axis at the zero.
• If even, the graph merely touches the x-axis and turns around at that point.

Recognizing the multiplicity of zeros helps us understand how the polynomial behaves in different regions of its graph.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. It is an indicator of the behavior and properties of the function. For example, the degree tells us how many roots the polynomial may have, including complex roots, and gives clues to the shape and end behavior of its graph.

In our exercise, the polynomial has a degree of 4. This is determined by adding the multiplicities of the zeros from the factorized form, \((x - 1)^2 * (x - 4)^2\), resulting in a total degree of 2 + 2 = 4. An even degree influences the end behavior and implies that the graph will always have the same direction at both ends.

End Behavior of Polynomials

End behavior refers to how the graph of a polynomial function behaves as \( x \) approaches positive or negative infinity. It is determined by the leading term and the degree of the polynomial. For polynomials:

• If the degree is even and the leading coefficient is positive, the ends of the graph point upwards.
• If the degree is even and the leading coefficient is negative, both ends will point downwards.
• If the degree is odd, the ends of the graph will point in opposite directions.

In the provided exercise, the graph of the polynomial falls to both the left and right, indicating the degree is even with a negative leading coefficient. This observation confirms the function’s end behavior, fitting perfectly with the polynomial \((x - 1)^2 * (x - 4)^2\) when fully expanded.