Problem 15 - Precalculus: Functions and Graphs

Question

Find the zeros of \(f(x),\) and state the multiplicity of each zero. $$f(x)=x^{2}(3 x+2)(2 x-5)^{3}$$

Step-by-Step Solution

Answer

The zeros are 0 (multiplicity 2), -2/3 (multiplicity 1), 5/2 (multiplicity 3).

Step 1: Identify Factors

The function is given as \( f(x) = x^{2}(3x+2)(2x-5)^{3} \). The factors are \( x^2 \), \( (3x + 2) \), and \( (2x - 5)^3 \).

Step 2: Solve for Zero of x²

Set the factor \( x^2 = 0 \). Solving for \( x \) gives \( x = 0 \). This occurs when the factor \( x^2 \) is equal to zero.

Step 3: Solve for Zero of (3x+2)

Set the factor \( 3x + 2 = 0 \). Solving for \( x \) gives \( 3x = -2 \) or \( x = -\frac{2}{3} \).

Step 4: Solve for Zero of (2x-5)

Set the factor \( (2x - 5)^3 = 0 \). Solving for \( x \) gives \( 2x = 5 \) or \( x = \frac{5}{2} \).

Step 5: Determine Multiplicities

The zero \( x = 0 \) comes from \( x^2 \) and has multiplicity 2. The zero \( x = -\frac{2}{3} \) comes from \( (3x+2) \) and has multiplicity 1. The zero \( x = \frac{5}{2} \) comes from \( (2x-5)^3 \) and has multiplicity 3.

Key Concepts

Zeros of Functions

When we talk about the "zeros of a function," we're referring to the values of the variable where the function equals zero. For a polynomial function, zeros are the solutions to the equation when set to zero, essentially the x-values where the graph intersects the x-axis.

In the given example, the polynomial function is expressed as a product of factors: \( f(x) = x^{2}(3x+2)(2x-5)^{3} \). Each factor can be individually set to zero to find the corresponding zeros:

• For \( x^2 \), setting \( x^2 = 0 \) gives the zero \( x = 0 \).
• For \( 3x + 2 \), setting \( 3x + 2 = 0 \) gives \( x = -\frac{2}{3} \).
• For \( (2x - 5)^3 \), setting \( (2x - 5)^3 = 0 \) gives \( x = \frac{5}{2} \).

These zeros are crucial for understanding the behavior of the polynomial function and determining where it cuts the x-axis.

Factorization

Factorizing a polynomial is breaking it down into simpler polynomials that multiply together to give the original polynomial. It's like taking apart a complex structure into its building blocks.

In the function \( f(x) = x^{2}(3x+2)(2x-5)^{3} \), we see that it consists of three factors: \( x^2 \), \( 3x+2 \), and \( (2x-5)^3 \). By factorizing, we express the polynomial function in terms of its component parts.

• Each polynomial factor is an expression that contributes roots to the polynomial.
• The power of each factor indicates how many times it is repeated, which affects the graph of the function.
• Factorization simplifies the process of solving polynomial equations by allowing us to evaluate each part individually.

Understanding factorization is essential for finding zeros and analyzing polynomials, as it breaks down complex expressions into manageable chunks.

Multiplicity of Roots

The multiplicity of a root refers to how many times a particular solution appears as a root of the polynomial. It provides insight into the shape of the graph at that intersection point with the x-axis.

For the polynomial \( f(x) = x^{2}(3x+2)(2x-5)^{3} \), we can determine the multiplicity by examining the exponents of the factors:

• The root \( x = 0 \) has multiplicity 2 because of \( x^2 \).
• The root \( x = -\frac{2}{3} \) has multiplicity 1 due to \( 3x + 2 \).
• The root \( x = \frac{5}{2} \) has multiplicity 3 because of \( (2x-5)^3 \).

The multiplicity affects the graph's behavior:

• Even multiplicities (like 2) cause the graph to touch and rebound off the x-axis.
• Odd multiplicities (like 3) allow the graph to cross the x-axis.

Recognizing root multiplicity is vital for predicting graph intersections and the nature of the polynomial's zeros.

Solving Polynomial Equations

Solving polynomial equations involves finding the values of the variable that make the equation true, or where the function equals zero. These solutions are identical to the "zeros" or "roots" of the function.

To solve the polynomial equation \( f(x) = x^{2}(3x+2)(2x-5)^{3} = 0 \), follow these steps:

• Identify each factor of the polynomial.
• Set each factor equal to zero and solve for the variable.
• Consider the multiplicity of each solution to understand its impact.

This approach ensures you capture all potential solutions:

• For \( x^2 = 0 \), \( x = 0 \) is a solution.
• For \( 3x + 2 = 0 \), \( x = -\frac{2}{3} \) is a solution.
• For \( (2x-5)^3 = 0 \), \( x = \frac{5}{2} \) is a solution.

By solving polynomial equations, we can fully explore the function's intercepts and learn about its graph.