91Ó°ÊÓ

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 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.

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.

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.