Problem 71 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. How many zeros, real and nonreal, does the function \(f(x)=x^{4}-1\) have? How many \(x\) -intercepts does the graph of \(f\) have?

Step-by-Step Solution

Answer

The function \(f(x)=x^{4}-1\) has four zeros: two real ones (\(-1\) and \(1\)) and two nonreal ones (\(i\) and \(-i\)). The graph of \(f\) has two x-intercepts, -1 and 1.

Step 1: Expression of the function

The given function is \(f(x)=x^{4}-1\). Set this function equal to zero to find the zeros (and x-intercepts). So, we would have \(x^{4} - 1 = 0\)

Step 2: Factor the equation

Next, factor the equation. The expression on the left side is a difference of squares, which factors into \((x^{2} - 1)(x^{2} + 1)\). Setting this equal to zero gives the equation \((x^{2} - 1)(x^{2} + 1) = 0\)

Step 3: Solving the factored equation

The solutions or zeros of the function are found by setting each factor equal to zero and solving for \(x\). This step will give the real zeros. Solving \(x^{2} - 1 = 0\) yields \(x = -1, 1\). Trying to solve \(x^{2} + 1 = 0\) for \(x\) will yield nonreal roots: \(x = i,-i\). Those are imaginary numbers and are not x-intercepts.

Step 4: Summary of the solution

In total, the function \(f(x)=x^{4}-1\) has four zeros: two real (which are also the x-intercepts) and two nonreal.

Key Concepts

Real and Nonreal Zeros

In the world of polynomials, zeros are the solutions or \(x\)-values that satisfy the equation when the polynomial is set to zero. These can be classified into real and nonreal zeros.

• Real Zeros: These are the solutions that are real numbers and can be visualized on the real number line. For the polynomial \(f(x)=x^4-1\), solving \(x^2-1=0\) leads us to the real zeros \(x = -1\) and \(x = 1\).
• Nonreal Zeros: These are not real numbers and generally involve complex numbers, featuring an imaginary component \(i\), where \(i^2 = -1\). For instance, solving \(x^2+1=0\) gives us the zeros \(x = i\) and \(x = -i\), which are nonreal.

Knowing the difference between real and nonreal zeros is essential as it helps us understand the behavior of polynomials, especially when visualizing their graphs.

X-Intercepts

The \(x\)-intercepts of a function's graph are the points where it crosses the \(x\)-axis. These intercepts are simply the real zeros of the polynomial function. In simpler terms, they are the values of \(x\) where the function equals zero.

For the function \(f(x)=x^{4}-1\), the \(x\)-intercepts are found by setting the function equal to zero and solving. After factoring, as calculated in the solution, \(x = -1\) and \(x = 1\) are the real zeros. These values are also the coordinates of the \(x\)-intercepts: \(-1, 0\) and \(1, 0\). Nonreal zeros such as \(x = i\) and \(x = -i\) do not appear as \(x\)-intercepts because they are not real numbers and cannot be plotted on the standard real-day graph.

Factoring Polynomials

Factoring polynomials is a method of breaking down an equation into simpler polynomials that can be multiplied together to get the original polynomial. It is especially useful when you need to find the zeros of a function.

• Difference of Squares: A difference of squares is a specific case where a square number is subtracted from another square number, such as \(a^2 - b^2\). This can be factored into \((a-b)(a+b)\). For the function \(f(x)=x^4-1\), we apply the difference of squares to get \((x^2-1)(x^2+1)\).
• Further Factoring: Once we factor \(x^4-1\) into \((x^2-1)(x^2+1)\), \(x^2-1\) can be further factored into \((x-1)(x+1)\). This exposes the real zeros that correspond with \(x\)-intercepts, as discussed earlier.

By understanding and applying factoring techniques, especially recognizing patterns like the difference of squares, finding the zeros of polynomials becomes more straightforward.