Problem 22 - College Algebra

Question

For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ f(x)=x^{6}-3 x^{4}-4 x^{2} $$

Step-by-Step Solution

Answer

The x-intercepts are \( x = 0, 2, -2 \).

Step 1: Set the Polynomial Function to Zero

To find the x-intercepts of the polynomial function, we need to set the function equal to zero and solve for x. So, start by setting \( f(x) = x^6 - 3x^4 - 4x^2 = 0 \).

Step 2: Factor Out the Greatest Common Factor

Notice that each term in the polynomial has a common factor of \( x^2 \). So factor \( x^2 \) from the polynomial: \( x^2 (x^4 - 3x^2 - 4) = 0 \).

Step 3: Apply Zero Product Property

According to the zero product property, if \( x^2 (x^4 - 3x^2 - 4) = 0 \), then either \( x^2 = 0 \) or \( x^4 - 3x^2 - 4 = 0 \). Solve each equation separately. Start by solving \( x^2 = 0 \).

Step 4: Solve for x from \( x^2 = 0 \)

Taking the square root of both sides, we find \( x = 0 \). So, one x-intercept is \( x = 0 \).

Step 5: Substitute \( x^2 = y \) in \( x^4 - 3x^2 - 4 = 0 \)

Let \( y = x^2 \). Then, the equation becomes a quadratic in terms of y: \( y^2 - 3y - 4 = 0 \).

Step 6: Solve the Quadratic Equation in y

Use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve \( y^2 - 3y - 4 = 0 \), where \( a = 1, b = -3, c = -4 \).

Step 7: Calculate the Discriminant

Calculate \( b^2 - 4ac = (-3)^2 - 4(1)(-4) = 9 + 16 = 25 \). Since the discriminant is positive, there are two real solutions.

Step 8: Solve for y

Using the quadratic formula, find the solutions: \( y_{1,2} = \frac{3 \pm \sqrt{25}}{2} = \frac{3 \pm 5}{2} \). These simplify to \( y_1 = 4 \) and \( y_2 = -1 \).

Step 9: Re-substitute and Solve for x

Recall \( y = x^2 \). Thus, \( x^2 = 4 \) or \( x^2 = -1 \). Since \( x^2 = -1 \) has no real solutions, only solve \( x^2 = 4 \), which gives \( x = 2 \) or \( x = -2 \).

Step 10: List all x-intercepts

Combine all solutions: \( x = 0, 2, -2 \). These are the x-intercepts of the function \( f(x) \).

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The most common form of a polynomial function in one variable is:

• It is expressed as: \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where each term can have a different power of \( x \) and \( a_n, a_{n-1}, ..., a_0 \) are coefficients.
• The degree of a polynomial is defined by the highest power of \( x \) with a non-zero coefficient, which determines the function's general shape and behavior.
• Polynomial functions can model a wide range of real-world phenomena due to their capacity to represent curves dictated by various powers of \( x \).

In the given problem, the function is \( f(x) = x^6 - 3x^4 - 4x^2 \), a degree 6 polynomial due to the term \( x^6 \). To find its x-intercepts, we set \( f(x) = 0 \) and solve for \( x \). This involves factoring and utilizing other algebraic techniques.

Zero Product Property

The zero product property is a crucial principle in algebra that helps in solving polynomial equations. It states:

• If a product of two or more factors equals zero, then at least one of the factors must be zero. Mathematically, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \).
• This property is particularly useful when dealing with factored forms of polynomial equations.

In the solution, after factoring the polynomial to get \( x^2(x^4 - 3x^2 - 4) = 0 \), the zero product property is applied such that either \( x^2 = 0 \) or \( x^4 - 3x^2 - 4 = 0 \). Solving these simpler equations gives the possible \( x \) values, identifying the x-intercepts of the polynomial function.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \). Key features of quadratic equations include:

• They have a degree of 2, indicated by the highest power of the variable being squared.
• The solutions to a quadratic equation can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• The expression inside the square root, \( b^2 - 4ac \), is called the discriminant, determining the nature the solutions.

For \( f(x) = x^6 - 3x^4 - 4x^2 \), we let \( y = x^2 \). This converts the polynomial \( x^4 - 3x^2 - 4 = 0 \) into the quadratic equation \( y^2 - 3y - 4 = 0 \), which can be solved for \( y \) using the quadratic formula. Thus, finding the values of \( y \), helps us further break down the solutions for \( x \).

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers or terms. In polynomial expressions:

• It is often used to simplify expressions and make solving polynomial equations easier by factoring them.
• With polynomials, look for common powers of variables or coefficients in each term.

In our problem \( f(x) = x^6 - 3x^4 - 4x^2 \), notice that each term contains \( x^2 \). By factoring out \( x^2 \), we simplify the polynomial to \( x^2(x^4 - 3x^2 - 4) = 0 \). Factoring reduces complexity, allowing you to solve for \( x \) by finding common elements. This technique ensures simpler expressions to apply the zero product property.