Problem 29 - College Algebra

Question

Find the intercepts of the functions. $$f(x)=x\left(x^{2}-2 x-8\right)$$

Step-by-Step Solution

Answer

The x-intercepts are \( x = 0, 4, -2 \) and the y-intercept is 0.

Step 1: Find the x-intercepts

To find the x-intercepts, set the equation \[ f(x) = 0 \].The equation becomes:\[ x(x^2 - 2x - 8) = 0 \].Since the product is zero, set each factor to zero: - \( x = 0 \). - \( x^2 - 2x - 8 = 0 \).Solve the quadratic equation \( x^2 - 2x - 8 = 0 \) next.

Step 2: Solve the Quadratic Equation

The quadratic equation is \( x^2 - 2x - 8 = 0 \). To solve it, factor the expression:\[ (x - 4)(x + 2) = 0 \].Set each factor equal to zero: - \( x - 4 = 0 \) gives \( x = 4 \). - \( x + 2 = 0 \) gives \( x = -2 \).Therefore, the x-intercepts are \( x = 0 \), \( x = 4 \), and \( x = -2 \).

Step 3: Find the y-intercept

To find the y-intercept, evaluate the function at \( x = 0 \):\[ f(0) = 0(0^2 - 2 \cdot 0 - 8) = 0 \].Thus, the y-intercept is 0.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that can be written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In the given exercise, we transformed the quadratic part into \( x^2 - 2x - 8 = 0 \). Quadratic equations are fundamental in algebra. They usually have two solutions, which could be real or complex numbers. These solutions are the values that make the entire equation zero.

• Real solutions happen where the graph intersects or touches the x-axis.
• Complex solutions occur when the graph does not intersect the x-axis at all.

Quadratic equations can often be solved by various methods like factoring, using the quadratic formula, or completing the square. In this problem, we chose factoring because it is convenient when the quadratic can be easily divided into two binomials.

Factoring

Factoring is a method that involves expressing a polynomial as the product of its factors, which can help solve equations or simplify expressions. For the quadratic equation \( x^2 - 2x - 8 = 0 \), we used factoring to find its solutions. To factor, look for two numbers that multiply to give \( -8 \) (the constant term) and add up to \( -2 \) (the coefficient of the linear term \( x \)).
The numbers \( -4 \) and \( +2 \) fit these criteria, so the factors are \((x - 4)\) and \((x + 2)\). By setting each factor equal to zero, \((x - 4)=0\) and \((x + 2)=0\), we find the solutions \( x = 4 \) and \( x = -2 \).
Factoring is often preferred because it provides a straightforward way to find solutions, especially when dealing with simple quadratic equations. Recognizing patterns or using techniques like grouping can make the factoring process easier.

Y-Intercepts

The y-intercept of a function is the point where the graph of the function intersects the y-axis, meaning at \( x = 0 \). To find it, simply substitute zero for \( x \) in the equation.
In our example, the function is given by \( f(x) = x(x^2 - 2x - 8) \). By substituting \( x = 0 \), we evaluate it as follows:

• \( f(0) = 0(0^2 - 2 \times 0 - 8) = 0 \).

This means the y-intercept is at the origin, or the point (0,0).
Identifying y-intercepts is crucial since it helps us understand the vertical positioning of a function’s graph and can aid in sketching the overall shape of the graph.

Solving Equations

Solving equations involves finding the value of variables that make the equation true. Solutions to equations can represent various real-world phenomena like times, distances, and amounts.
In our exercise, we tackled both linear and quadratic equations. Initially, after setting \( f(x) = 0 \), we solve \( x(x^2 - 2x - 8) = 0 \) by isolating factors that multiply to zero. This is based on the Zero Product Property, which states that if a product of factors equals zero, then at least one of the factors must be zero.

• \( x = 0 \)
• \( x^2 - 2x - 8 = 0 \)

After factoring \( x^2 - 2x - 8 \) into \( (x - 4)(x + 2) \), we found the solutions: \( x = 4 \) and \( x = -2 \).
Finally, these solutions represent the x-intercepts of the graph. Solving equations not only helps in graphical interpretations but also in understanding mathematical relationships between various quantities.