Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all the zeros of the function. Is there a relationship between the number of real zeros and the number of \(x\) -intercepts of the graph? Explain. $$\begin{aligned} f(x)=& x^{3}-4 x^{2} \\ &-4 x+16 \end{aligned}$$

Step-by-Step Solution

Answer

The zeros of the function \(f(x) = x^3 - 4x^2 - 4x +16\) are \(x = 2\), \(x = -2\), and \(x = -4\). These zeros correspond to the \(x\)-intercepts of the function graph, showing a direct relationship between the zeros and \(x\)-intercepts.

Step 1: Setting the function to zero

First, to find the zeros of \(f(x)\), set \(f(x)\) to zero, so you will have an equation to solve: \(x^3 - 4x^2 - 4x +16 = 0\)

Step 2: Solving the equation

Next, solve the equation for \(x\). This is a cubic equation. There aren't any simple methods for solving such equation analytically, but it can be solved with a computer or a calculator. The roots for this equation are \(x = 2\), \(x = -2\), and \(x = -4\).

Step 3: Analyzing the relationship between zeros and \(x\)-intercepts

In single-variable calculus, a zero of a function is a point at which the function evaluates to 0. On the function graph, this value of \(x\) corresponds to a point on the \(x\)-axis, known as the \(x\)-intercept. Therefore, the zeros of a function and the \(x\)-intercepts of the graph of a function in one variable are the same thing. In other words, there is a direct relationship between the zeros of a function and the \(x\)-intercepts of a function graph

Step 4: Connecting the results with the function graph

The function \(f(x) = x^3 - 4x^2 - 4x +16\) is a cubic function. It crosses the \(x\)-axis at its real zeros, which are \(x = 2\), \(x = -2\), and \(x = -4\). Thus, the number of \(x\)-intercepts of this function graph corresponds to the number of real zeros of the function. In our case it is 3.

Key Concepts

Cubic Functions

Cubic functions are a type of polynomial equation with a degree of three. They are represented by the general form: \[ f(x) = ax^3 + bx^2 + cx + d \] where \( a \), \( b \), \( c \), and \( d \) are constants, and \( a eq 0 \). These functions can have up to three real roots or zeros, which are the values of \( x \) that solve the equation \( f(x) = 0 \).

• They can cross the x-axis up to three times, which is a unique characteristic among polynomial functions.
• The graph of a cubic function can show various behaviors, including one or two turns, known as inflection points.
• Since they're not limited to linear characteristics, cubic functions are powerful in modeling real-world situations where change is not constant.

Consider the given equation: \[ f(x) = x^3 - 4x^2 - 4x + 16 \]. It's clearly a cubic equation because the highest exponent of \( x \) is 3. Understanding these basic properties helps in approaching the zeros systematically.

X-Intercepts

The x-intercepts of a function graph are the points where the function crosses or touches the x-axis. To find these intercepts in the context of a function like \( f(x) = x^3 - 4x^2 - 4x + 16 \), we must determine where the function equals zero:The process involves: - Setting the equation to zero: \[ x^3 - 4x^2 - 4x + 16 = 0 \]- Solving this equation will give us the x-values (or roots) which are the x-intercepts.In this exercise, the computed x-intercepts are \( x = 2 \), \( x = -2 \), and \( x = -4 \). Each of these points represents a solution where the function graph meets the x-axis. Finding x-intercepts is crucial as they offer insights into the real roots of the polynomial and provide the breaking points on the function's graph.

Roots of Equation

The roots of a polynomial equation, such as a cubic equation, are the values of \( x \) that make the function equal to zero. These are also known as "zeros," indicating where the graph hits the x-axis. In our equation, \( x^3 - 4x^2 - 4x + 16 = 0 \), the roots are calculated to be \( x = 2 \), \( x = -2 \), and \( x = -4 \).These roots also serve to:

• Provide the specific x-values where the graph intersects the axis, reflecting the turning points or crucial transitions in the behavior of the graph.
• Act as keys in sketching the graph of the polynomial, allowing predictions on how the function behaves within intervals formed by these roots.
• Validate algebraic solutions with visual understanding through graph diagrams and potential real-life applications where determining exact crossing points is necessary.

By identifying these roots, students can check and verify the accuracy of solving equations and understand the deeper connections between algebraic expressions and their graphical representations.