Problem 65 - Algebra for College Students

Question

Find all real zeros to each polynomial function by graphing the function and locating the \(x\) -intercepts. $$ f(x)=x^{3}-0.2 x^{2}-0.05 x+0.006 $$

Step-by-Step Solution

Answer

The real zeros are approximately x ≈ -0.1, 0.06, and 0.2.

Step 1 - Understand the Function

First, take a look at the given polynomial function. The function is \[f(x) = x^3 - 0.2x^2 - 0.05x + 0.006\]This is a cubic polynomial, meaning it can have up to three real zeros.

Step 2 - Graph the Function

Use graphing technology or graphing calculator to graph the function \(f(x) = x^3 - 0.2x^2 - 0.05x + 0.006\).

Step 3 - Identify the x-Intercepts

Look at where the graph crosses the x-axis. These points are the real zeros of the function.

Step 4 - Record the Zeros

Read off the x-coordinates of the points where the graph crosses the x-axis. These are the solutions to the polynomial equation.

Key Concepts

graphing polynomial functions

Graphing polynomial functions is an important skill for understanding their behavior. A polynomial function is a mathematical expression made up of terms combined by addition, subtraction, and multiplication. The general form is given by: \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \text{...} + a_1x + a_0 \] Using graphing technology or a graphing calculator helps visualize the function's shape. By plotting the polynomial, you can easily identify important features, such as turning points, end behavior, and the points where it crosses the axes. For the function \( f(x) = x^3 - 0.2x^2 - 0.05x + 0.006 \), graphing will reveal insights that are not immediately apparent by just looking at the equation.

x-intercepts

The x-intercepts of a polynomial function are the points where the graph crosses the x-axis. These intercepts represent the real zeros of the function, or the solutions to the equation \( f(x) = 0 \). To find the x-intercepts of \( f(x) = x^3 - 0.2x^2 - 0.05x + 0.006 \):

• Graph the function using graphing technology.
• Identify the points where the graph meets the x-axis.
• Record the x-coordinates of these points; they are your real zeros.

For example, if the graph crosses the x-axis at \( x = 0.1 \), \( x = 0.2 \), and \( x = -0.1 \), these values are the real zeros of the polynomial.

cubic polynomials

Cubic polynomials are a special type of polynomial function of degree 3. They have the general form: \[ f(x) = ax^3 + bx^2 + cx + d \] where \( a eq 0 \). Cubic polynomials have several key characteristics:

• They can have up to three real zeros.
• The graph of a cubic polynomial can have one or two turning points.
• The end behavior of the function depends on the leading coefficient \( a \): if \( a > 0 \), the graph rises to the right and falls to the left; if \( a < 0 \), the graph rises to the left and falls to the right.

For the function \( f(x) = x^3 - 0.2x^2 - 0.05x + 0.006 \), understanding that it is a cubic polynomial helps predict its behavior and the number of possible real zeros.

polynomial zeros

Polynomial zeros are the solutions to the polynomial equation \( f(x) = 0 \). These zeros can be real or complex, but the task often focuses on identifying the real zeros. Different methods can be used to find the zeros:

• Graphing the polynomial and identifying the x-intercepts.
• Using algebraic methods such as factoring, synthetic division, or the Rational Root Theorem.
• Applying numerical methods if the algebraic methods are too complex.

For \( f(x) = x^3 - 0.2x^2 - 0.05x + 0.006 \), graphing is a convenient approach. By plotting the function, you can visually identify where the graph crosses the x-axis, giving you the real zeros of the polynomial.