Question

At what \(x\) -values does the function \(y=x(x-5)(x+2)\) intersect the \(x\)-axis? (A) \(-10\) (B) \(0,3,12\) (C) \(2,-5\) (D) \(0,5,-2\)

Short answer

The function intersects the x-axis at options (D) 0, 5, -2.

Step-by-step solution

Understanding the Function and the x-axis

The function given is in the form of a product of factors, that is, \( y = x(x-5)(x+2) \). For the function to intersect the \(x\)-axis, \(y\) must equal zero at those points. Therefore, to find the intersection points with the \(x\)-axis, solve for \(x\) where \( y = 0 \).

Setting the Function Equal to Zero

Set the equation \( y = x(x-5)(x+2) \) equal to zero: \[x(x-5)(x+2) = 0\] This equation will be zero if any of the factors \(x\), \(x-5\), or \(x+2\) is zero.

Solving for x-values

Solve each factor individually:1. \(x = 0\) 2. \(x - 5 = 0 \Rightarrow x = 5\) 3. \(x + 2 = 0 \Rightarrow x = -2\)Thus, the \(x\)-values where the function intersects the \(x\)-axis are \(x = 0\), \(x = 5\), and \(x = -2\).

Choosing the Correct Answer

From the solutions \(x = 0\), \(x = 5\), and \(x = -2\), we can see that the correct answer matches option (D) 0, 5, -2.

Key concepts

Understanding Polynomial Equations

A polynomial equation is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomial equations like the one in our original exercise, blend multiple terms including constants and variables to determine a specific outcome. In the function provided, \( y = x(x-5)(x+2) \), we can observe that it is a polynomial of degree 3. This implies it is a cubic polynomial because the highest power of \(x\) is 3. Understanding polynomial equations is critical because they form the basis for mathematical modeling in diverse fields, from physics to economics.

• The degree of a polynomial indicates the number of roots or solutions it might have.
• Each factor of a polynomial can provide up to one solution if set to zero.
• Polynomial equations can frequently be factored into simpler binomials or monomials.

To solve polynomial equations, it helps to express them in a factored form, which simplifies the process of finding solutions.

Finding X-Intercepts

The x-intercepts of a function are the points where the graph touches or crosses the x-axis. At these points, the function’s output, \(y\), is zero. For polynomial functions, this involves identifying what values of \(x\) make the function equal to zero. In our exercise, the expression \( y = x(x-5)(x+2) \) was set to zero to find these points.
To find x-intercepts:

• Transform the function into a product of linear factors, if possible, as we have already done in this exercise.
• Identify each factor within the polynomial expression.
• Set each factor equal to zero and solve for \(x\).

Setting each factor of \(x(x-5)(x+2) = 0\) highlights that the x-intercepts for this function are at \(x = 0, 5,\) and \(-2\). These values are where the graph cuts or touches the x-axis, indicating they represent the roots where the function equals zero.

Exploring Roots of Equations

The roots of an equation are the solution values that satisfy the equation when it is set to zero. These are also known as "solutions" or "zeroes" of the equation. For the polynomial given in our exercise, the roots are the x-values where the function \( y = x(x-5)(x+2) \) equals zero. Let's explore some key points about roots:

• Roots are found by solving the polynomial equation set to zero.
• Multiple roots can exist. In a third-degree polynomial like this, there are up to three roots.
• Real roots are those that occur on the x-axis, while complex roots will not intersect the x-axis and involve imaginary numbers.

In this example, we found the real roots by factoring the polynomial. After setting each factor to zero, the solutions \(x = 0, 5, \) and \(-2\) give us the roots. In essence, identifying the roots of equations allows us to understand the behavior of polynomial functions and predict their interactions with the x-axis effectively.