Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=2 x^{4}\left(x^{3}-4 x^{2}+4 x\right) $$

Step-by-Step Solution

Answer

Zeros are 0 (multiplicity 5) and 2 (multiplicity 2).

Step 1: Set the function equal to zero

To find the zeros of the function, we start by setting \( f(x) = 2x^4(x^3 - 4x^2 + 4x) = 0 \). This gives us an equation to solve.

Step 2: Factor the equation

Factor the expression \( x^3 - 4x^2 + 4x \). First, factor out an \( x \):\[x(x^2 - 4x + 4)\]Next, notice that \( x^2 - 4x + 4 \) is a perfect square:\[x(x-2)^2\]Thus, the function becomes \( 2x^4x(x-2)^2 = 0 \).

Step 3: Solve for zeros

The factored equation is \( 2x^5(x-2)^2 = 0 \). Solve for \( x \) by setting each factor to zero:1. \( 2x^5 = 0 \): This reduces to \( x=0 \).2. \( (x-2)^2 = 0 \): This gives \( x=2 \).These are the zeros of the function.

Step 4: Determine the multiplicity of each zero

The zero \( x=0 \) comes from the factor \( x^5 \), so it has a multiplicity of 5. The zero \( x=2 \) comes from the factor \((x-2)^2\), so it has a multiplicity of 2.

Key Concepts

Zeros of Functions

Understanding the zeros of functions is crucial when analyzing polynomial equations. Zeros, often referred to as "roots," are the points where the function crosses or touches the x-axis. Essentially, these are the x-values for which the function equals zero. Let's break it down with simple steps:

• Set the polynomial equation equal to zero to find these points.
• The solutions to the equation are the zeros of the polynomial function.

For example, in our exercise, the function is set as follows:
\( f(x) = 2x^4(x^3 - 4x^2 + 4x) = 0 \).
By solving this, we identify the zeros of the function, which are crucial for graphing and understanding the behavior of polynomial functions.

Factorization

Factorization is a method used to simplify polynomial equations and find their zeros more easily. By expressing the polynomial as a product of its factors, we can set each factor equal to zero to find the solutions. Here's how factorization works:

• Identify common factors and take them out of the equation. This often simplifies the equation significantly.
• Look for other patterns, like perfect squares or difference of squares, that allow further factoring.

In the exercise, we began with
\( x^3 - 4x^2 + 4x \),
and factored it as follows:
\[x(x-2)^2\]Now, the polynomial is completely factored as
\( 2x^4x(x-2)^2 \).
This makes it much easier to identify zeros.

Multiplicity of Zeros

Multiplicity refers to how many times each zero appears in the factorization of the polynomial. It gives us insight into the behavior of the graph at each zero. When a factor is repeated, it affects the shape of the graph and how it touches or crosses the x-axis. Here’s what to consider:

• A zero with even multiplicity, like \( x = 2 \) with multiplicity of 2, implies the graph touches the x-axis and turns around.
• A zero with odd multiplicity, like \( x = 0 \) with multiplicity of 5, tends to cross the x-axis.

In our exercise, after factoring, we identified:
- Zero \( x = 0 \) (from \( x^5 \)) with multiplicity of 5.
- Zero \( x = 2 \) (from \( (x-2)^2 \)) with multiplicity of 2.
Multiplicity is key to predicting the graph's local behavior at each zero, thereby enriching our understanding of polynomial functions.