Finding X-Axis Intercepts Of Polynomials A Step-by-Step Solution

Jul 10, 2025 by ADMIN 65 views

Understanding X-Axis Intercepts of Polynomial Functions

Polynomial functions are fundamental in mathematics, and understanding their behavior is crucial for various applications. One key aspect of analyzing polynomial functions is identifying their x-axis intercepts, also known as roots or zeros. These are the points where the graph of the function crosses the x-axis, and they provide valuable information about the function's solutions and behavior. In this article, we will delve into the process of finding x-axis intercepts, focusing on the given polynomial function and the methods to determine its roots. We'll explore the concept of factoring polynomials, the zero-product property, and how these tools help us pinpoint the exact points where the graph intersects the x-axis. By the end of this exploration, you'll have a solid understanding of how to analyze polynomial functions and extract meaningful information from their x-axis intercepts.

The x-axis intercepts of a function are the points where the graph of the function intersects the x-axis. At these points, the y-coordinate is always zero. Therefore, to find the x-axis intercepts of a function, we need to solve the equation y = 0. In the context of a polynomial function, this means finding the values of x that make the polynomial equal to zero. These values are also known as the roots or zeros of the polynomial.

To find the x-axis intercepts of the given polynomial function, $y=(x-5)(x^2-7x+12)$ , we set y equal to zero and solve for x:

$(x-5)(x^2-7x+12) = 0$

This equation tells us that the product of two factors, $(x-5)$ and $(x^2-7x+12)$ , is equal to zero. To solve this, we can use the zero-product property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. Applying this property, we get two separate equations:

$x - 5 = 0$
$x^2 - 7x + 12 = 0$

Solving the first equation is straightforward:

$x - 5 = 0$

Add 5 to both sides:

$x = 5$

So, one of the x-axis intercepts is at $x = 5$ . This corresponds to the point $(5, 0)$ on the graph.

The second equation is a quadratic equation. To solve it, we can try to factor the quadratic expression:

$x^2 - 7x + 12 = 0$

We are looking for two numbers that multiply to 12 and add up to -7. These numbers are -3 and -4. Therefore, we can factor the quadratic as follows:

$(x - 3)(x - 4) = 0$

Now, we apply the zero-product property again:

$x - 3 = 0$
$x - 4 = 0$

Solving these equations:

For $x - 3 = 0$ , add 3 to both sides:

$x = 3$

So, another x-axis intercept is at $x = 3$ , which corresponds to the point $(3, 0)$ on the graph.

For $x - 4 = 0$ , add 4 to both sides:

$x = 4$

This gives us another x-axis intercept at $x = 4$ , corresponding to the point $(4, 0)$ on the graph.

Therefore, the x-axis intercepts of the polynomial function $y = (x-5)(x^2-7x+12)$ are at the points $(3, 0)$ , $(4, 0)$ , and $(5, 0)$ . These are the points where the graph of the function crosses the x-axis. Understanding how to find these intercepts is crucial for sketching the graph of the polynomial and analyzing its behavior.

To find the x-axis intercepts of the graph of the equation $y=(x-5)(x^2-7x+12)$ , we need to determine the points where the graph intersects the x-axis. These points occur when $y = 0$ . Thus, we need to solve the equation:

$(x-5)(x^2-7x+12) = 0$

The given equation is a polynomial equation, and we can find its roots by setting each factor equal to zero. The equation is already partially factored, which simplifies the process. The first factor is $(x-5)$ , and the second factor is a quadratic expression $(x^2-7x+12)$ .

Step 1: Set the equation to zero

The equation is already set to zero:

$(x-5)(x^2-7x+12) = 0$

Step 2: Factor the quadratic expression

We need to factor the quadratic expression $x^2-7x+12$ . We are looking for two numbers that multiply to 12 and add up to -7. These numbers are -3 and -4. Thus, we can factor the quadratic as:

$x^2-7x+12 = (x-3)(x-4)$

So, the original equation becomes:

$(x-5)(x-3)(x-4) = 0$

Step 3: Apply the Zero Product Property The Zero Product Property states that if the product of several factors is zero, then at least one of the factors must be zero. Applying this property, we set each factor equal to zero:

$x-5 = 0$
$x-3 = 0$
$x-4 = 0$

Step 4: Solve each equation for x

Solve $x-5 = 0$ :

Add 5 to both sides:

$x = 5$

This gives us the point $(5, 0)$ .
Solve $x-3 = 0$ :

Add 3 to both sides:

$x = 3$

This gives us the point $(3, 0)$ .
Solve $x-4 = 0$ :

Add 4 to both sides:

$x = 4$

This gives us the point $(4, 0)$ .

Step 5: Identify the correct answer

From the solutions we found, the points where the graph crosses the x-axis are $(5, 0)$ , $(3, 0)$ , and $(4, 0)$ . Now, let's check which of the given options matches one of these points:

A. $(-5,0)$ : This point does not match any of our solutions.
B. $(-3,0)$ : This point does not match any of our solutions.
C. $(4,0)$ : This point matches one of our solutions.
D. $(12,0)$ : This point does not match any of our solutions.

Therefore, the correct answer is C. $(4,0)$ .

Option C: (4, 0) is the correct answer because when we substitute $x = 4$ into the equation $y=(x-5)(x^2-7x+12)$ , we get $y = 0$ . Let's verify this:

$y = (4-5)(4^2-7(4)+12)$

$y = (-1)(16-28+12)$

$y = (-1)(28-28)$

$y = (-1)(0)$

$y = 0$

Since $y = 0$ when $x = 4$ , the point $(4, 0)$ is indeed an x-axis intercept of the graph. This confirms our solution and demonstrates the importance of verifying the roots by substituting them back into the original equation.

To further solidify our understanding, let's analyze why the other options are incorrect:

Option A: (-5, 0)

Substitute $x = -5$ into the equation:

$y = (-5-5)((-5)^2-7(-5)+12)$

$y = (-10)(25+35+12)$

$y = (-10)(72)$

$y = -720$

Since $y eq 0$ , the point $(-5, 0)$ is not an x-axis intercept.
Option B: (-3, 0)

Substitute $x = -3$ into the equation:

$y = (-3-5)((-3)^2-7(-3)+12)$

$y = (-8)(9+21+12)$

$y = (-8)(42)$

$y = -336$

Since $y eq 0$ , the point $(-3, 0)$ is not an x-axis intercept.
Option D: (12, 0)

Substitute $x = 12$ into the equation:

$y = (12-5)(12^2-7(12)+12)$

$y = (7)(144-84+12)$

$y = (7)(72)$

$y = 504$

Since $y eq 0$ , the point $(12, 0)$ is not an x-axis intercept.

By analyzing these options, we clearly see that only option C, $(4, 0)$ , satisfies the condition $y = 0$ , making it the correct x-axis intercept.

In conclusion, finding the x-axis intercepts of a polynomial function involves setting the function equal to zero and solving for x. The zero-product property is a crucial tool in this process, especially when the polynomial can be factored. By factoring the polynomial $y=(x-5)(x^2-7x+12)$ , we identified the roots as $x = 3$ , $x = 4$ , and $x = 5$ , corresponding to the x-axis intercepts $(3, 0)$ , $(4, 0)$ , and $(5, 0)$ . The correct answer among the given options was C. $(4, 0)$ .

Understanding how to find x-axis intercepts is fundamental for sketching graphs of polynomial functions and analyzing their behavior. This skill is essential in various areas of mathematics and has practical applications in fields like engineering, physics, and economics. By mastering these techniques, you'll be well-equipped to tackle more complex problems involving polynomial functions and their graphical representations. Remember, the key is to set the function to zero, factor if possible, and apply the zero-product property to find the roots. Always verify your solutions by substituting them back into the original equation to ensure accuracy. This thorough approach will help you confidently identify x-axis intercepts and gain a deeper understanding of polynomial functions.