Problem 14 - College Algebra

Question

For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ f(x)=x^{3}+6 x^{2}-7 x $$

Step-by-Step Solution

Answer

The x-intercepts are \( x = 0 \), \( x = 1 \), and \( x = -7 \).

Step 1: Understand the problem

We need to find the x-intercepts of the polynomial function \( f(x) = x^{3} + 6x^{2} - 7x \). X-intercepts occur where \( f(x) = 0 \). So we must solve the equation \( x^{3} + 6x^{2} - 7x = 0 \) to find the values of \( x \).

Step 2: Factor the polynomial

Notice that each term in \( x^{3} + 6x^{2} - 7x \) contains \( x \), so we can factor \( x \) out of the polynomial. This gives us \( x(x^{2} + 6x - 7) = 0 \).

Step 3: Solve for x

Once factored, the equation \( x(x^{2} + 6x - 7) = 0 \) shows that a solution occurs either when \( x = 0 \) or when \( x^{2} + 6x - 7 = 0 \).

Step 4: Solve the quadratic equation

To solve \( x^{2} + 6x - 7 = 0 \), we can use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 6 \), and \( c = -7 \). Calculate the discriminant: \( b^2 - 4ac = 6^2 - 4(1)(-7) = 36 + 28 = 64 \). The discriminant is 64, so there are two real solutions.

Step 5: Apply the quadratic formula

Using the quadratic formula, \( x = \frac{-6 \pm \sqrt{64}}{2 \cdot 1} \). Simplifying gives two solutions: \( x = \frac{-6 + 8}{2} = 1 \) and \( x = \frac{-6 - 8}{2} = -7 \).

Step 6: Compile the x-intercepts

The x-intercepts of the polynomial function are \( x = 0 \), \( x = 1 \), and \( x = -7 \).

Key Concepts

x-intercepts

To understand x-intercepts, think of them as points where the graph of a polynomial function crosses the x-axis. In simpler terms, it's where the output of the function is zero. For any function \( f(x) \), the x-intercepts occur where the function is set to zero, i.e., \( f(x) = 0 \). These points are crucial since they can provide insights into the behavior of polynomial functions and aid in graphing them. Here's a quick way to identify them:

• Set the polynomial equation equal to zero.
• Solve for \( x \) to find where the function value reaches zero.

In our exercise, the x-intercepts of the function \( f(x) = x^3 + 6x^2 - 7x \) were determined by solving \( x^3 + 6x^2 - 7x = 0 \). This gave us the intercepts \( x = 0 \), \( x = 1 \), and \( x = -7 \).

factoring

Factoring is a powerful algebraic method used to simplify polynomial equations and find their solutions. When you factor a polynomial, you're breaking it down into simpler expressions (or factors) that multiply together to give the original polynomial.
In the context of finding x-intercepts, factoring helps identify solutions where the product is zero. For example, if \( ab = 0 \), either \( a = 0 \) or \( b = 0 \), or both, need to be true for the equation to hold.In our exercise, we factor the polynomial \( f(x) = x^3 + 6x^2 - 7x \) by noting that each term contains \( x \). Thus, we factor out \( x \) from every term:

• The original polynomial becomes \( x(x^2 + 6x - 7) = 0 \).
• Solving for \( x \) now involves solving \( x = 0 \) or \( x^2 + 6x - 7 = 0 \).

Factoring is a helpful first step to simplify and solve polynomial equations.

quadratic formula

The quadratic formula is an essential tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It gives us the exact roots of the equation, even when the quadratic does not factor easily. The formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula derives from completing the square and is particularly useful for complicated quadratics.
Steps to use the quadratic formula:

• Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation.
• Calculate the discriminant \( b^2 - 4ac \).
• Insert values into the formula and simplify.

In our exercise, the quadratic \( x^2 + 6x - 7 = 0 \) was solved using this formula, resulting in the roots \( x = 1 \) and \( x = -7 \). These solutions correspond to the x-intercepts of the polynomial function.

discriminant in quadratic equations

The discriminant is a component of the quadratic formula that helps predict the nature of the roots without actually solving the equation. It is represented as \( \Delta = b^2 - 4ac \).Here's what the discriminant reveals:

• If \( \Delta > 0 \), the quadratic has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root (a repeated root).
• If \( \Delta < 0 \), the roots are complex and not real.

In the step-by-step solution from our exercise, the discriminant for \( x^2 + 6x - 7 = 0 \) was calculated as 64. Since 64 is greater than zero, the quadratic equation has two real and distinct solutions. Understanding the discriminant helps anticipate the solutions' form, reducing surprising complexities when solving quadratics. This foresight is quite handy in both academic and real-world applications.