Problem 41 - Precalculus : Building Concepts and Connections

Question

Find an expression for a polynomial function \(f(x)\) having the given properties. There can be more than one correct anstoer. Degree \(4 ;\) zeros 2 and \(4,\) cach of multiplicity 2

Step-by-Step Solution

Answer

The expression for the polynomial function \(f(x)\) with degree 4 and zeros 2 and 4, each of multiplicity 2, is \(f(x) = x^4 - 12x^3 + 48x^2 - 64x + 64\).

Step 1: Express the zeros as factors

Express each zero as the factor \((x - zero)^{multiplicity}\). In this case, the zeros are 2 and 4, each with a multiplicity of 2. So, the factors would be \((x - 2)^2\) and \((x - 4)^2\).

Step 2: Combine the factors into a polynomial

To find the expression for the polynomial function, combine the factors found in step 1 by multiplying them together. The multiplication of these factors gives the required polynomial. So the polynomial function \(f(x)\) will be \(f(x) = (x - 2)^2 * (x - 4)^2\).

Step 3: Expand the polynomial

Expand the polynomial function, \(f(x)\), to get its final form. The polynomial \(f(x) = (x - 2)^2 * (x - 4)^2\), when expanded, will give the final polynomial function. The simplified form of \(f(x)\) is \(f(x) = x^4 - 12x^3 + 48x^2 - 64x + 64\).

Key Concepts

Zero Multiplicity

In polynomial functions, zeros play a crucial role, not just by their values but also by how many times they appear, known as their multiplicities. When a zero, say 2, is said to have a multiplicity of 2, it means the factor associated with this zero appears twice in the polynomial's factorization:

• This creates the factor \( (x - 2)^2 \).
• If you have another zero, say 4, with the same multiplicity, it becomes \( (x - 4)^2 \).

These multiplicities affect the shape of the graph. For example, zeros with even multiplicities like in this exercise will "bounce off" the x-axis. Understanding multiplicity helps in sketching the graph of the polynomial.

Polynomial Degree

The degree of a polynomial function is determined by the highest exponent in its expanded form. It tells you the maximum number of turning points and the behavior of the graph at the ends.

• In our exercise, the polynomial has zeros 2 and 4, both with multiplicity 2.
• The degree is calculated by adding up these multiplicities: \( 2 + 2 = 4 \).

This confirms the given degree of the polynomial as 4. Knowing the degree helps in predicting the graph’s shape and the number of possible x-intercepts.

Expanding Polynomials

Expanding a polynomial involves distributing and combining like terms to simplify the expression. For the given expression \( f(x) = (x - 2)^2 * (x - 4)^2 \):

• First, expand each squared term: \( (x - 2)^2 = x^2 - 4x + 4 \) and \( (x - 4)^2 = x^2 - 8x + 16 \).
• Next, multiply these expanded forms together using distribution: \( (x^2 - 4x + 4)(x^2 - 8x + 16) \).
• After distributing and combining like terms, you get \( f(x) = x^4 - 12x^3 + 48x^2 - 64x + 64 \).

Expanding makes it easier to find exact values and plot the polynomial.

Factoring Polynomials

Factoring is the reverse process of expanding. It involves breaking down a polynomial into simpler "factor" components, usually to find zeros or simplify expressions. Starting with an already factored polynomial: \( f(x) = (x - 2)^2 * (x - 4)^2 \):

• Notice the repeated factors corresponding to particular zeros, like \( (x - 2) \) and \( (x - 4) \).
• This factor form is particularly useful for identifying the zeros directly: 2 and 4.

Factoring makes it simple to solve equations set to zero or to focus on specific properties of polynomials, like their zeros and multiplicities.