Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - or t-intercepts of the polynomial functions. $$ f(x)=x^{4}-x^{2} $$

Step-by-Step Solution

Answer

The x-intercepts are x = 0, x = -1, and x = 1.

Step 1: Identify x-intercepts

The x-intercepts of a function are the values of \(x\) where the function equals zero. To find the x-intercepts of \(f(x) = x^4 - x^2\), we need to set the equation equal to zero: \(x^4 - x^2 = 0\).

Step 2: Factor the equation

Recognize that \(x^4 - x^2\) can be factored by taking out a common \(x^2\): \(x^2(x^2 - 1) = 0\).

Step 3: Factor further

The expression \(x^2 - 1\) is a difference of squares, which can be further factored as \((x+1)(x-1)\). Thus, our equation becomes \(x^2(x+1)(x-1) = 0\).

Step 4: Solve for x

Using the zero-product property, set each factor equal to zero: \(x^2 = 0\), \(x+1 = 0\), and \(x-1 = 0\). Solve these equations to find \(x = 0\), \(x = -1\), and \(x = 1\).

Key Concepts

x-intercepts

The concept of x-intercepts is really all about determining where a graph of a polynomial function crosses the x-axis. In simpler terms, this happens when the y-value is zero. So, to find the x-intercepts of a polynomial function like \( f(x) = x^4 - x^2 \), you need to identify the values of \( x \) that make the function equal to zero.

When looking at polynomial equations, setting the equation equal to zero is your starting line. For the given function, we set \( x^4 - x^2 = 0 \). These values of \( x \) give us the points, or intercepts, where the graph would touch or cross the x-axis. Finding these zeros is essential in plotting the polynomial on a graph and understanding its behavior.

factoring polynomials

Factoring polynomials is a crucial step in solving polynomial equations, especially when finding x-intercepts. The idea behind factoring is to break down a complex polynomial into simpler, easier-to-handle pieces called factors.

In the function \( f(x) = x^4 - x^2 \), the first step is to look for a common factor. Here, we can factor out \( x^2 \) because it appears in both terms. So the equation becomes \( x^2(x^2 - 1) = 0 \).

But we aren't done yet. The expression \( x^2 - 1 \) can be recognized as a difference of squares—a special type of polynomial that can be factored into two binomials: \( (x+1)(x-1) \). So, the factored form of the entire expression is \( x^2(x+1)(x-1) = 0 \).

This factored form is much more manageable and illustrates the roots of the polynomial clearly, which are the x-intercepts of the function.

zero-product property

The zero-product property is a handy rule in algebra that helps solve equations like our factored polynomial. This property states that if the product of several factors equals zero, then at least one of the factors must be zero.

In the equation \( x^2(x+1)(x-1) = 0 \), we apply this principle to find the solutions.

• First, consider \( x^2 = 0 \). The solution is \( x = 0 \).
• Next, solve \( x+1 = 0 \), giving the solution \( x = -1 \).
• Finally, for \( x-1 = 0 \), the solution comes out as \( x = 1 \).

These solutions, \( x = 0, -1, \) and \( 1 \), are the x-intercepts of the polynomial function. The zero-product property simplifies finding these points because it reduces the problem to solving linear equations for each factor that results in a zero product.