Using the Zero Product Property

Oct 13, 2025 by magicalceylontours.com 116 views

Introduction

When dealing with polynomial functions, it's essential to understand how to find the x-intercepts, which are the points where the graph of the function crosses the x-axis. In this article, we will explore how to find the x-intercept of the given polynomial function $$y=(x-5)\left(x^2-7x+12\right)$$.

Understanding the x-Intercept

The x-intercept of a function is the point where the graph of the function crosses the x-axis. At this point, the y-coordinate is always 0. In other words, the x-intercept is the solution to the equation $$f(x) = 0$$, where $$f(x)$$ is the given function.

Finding the x-Intercept of the Given Function

To find the x-intercept of the given function $$y=(x-5)\left(x^2-7x+12\right)$$, we need to set the function equal to 0 and solve for x.

y=(x-5)\left(x^2-7x+12\right) = 0

Using the Zero Product Property

The zero product property states that if the product of two or more factors is equal to 0, then at least one of the factors must be equal to 0. We can use this property to solve for x.

(x-5)\left(x^2-7x+12\right) = 0

This means that either $$(x-5) = 0$$ or $$(x^2-7x+12) = 0$$.

Solving for x

Let's solve for x in both cases.

Case 1: (x-5) = 0

(x-5) = 0

x = 5

Case 2: (x^2-7x+12) = 0

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

We can factor the quadratic expression as:

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

This means that either $$(x-3) = 0$$ or $$(x-4) = 0$$.

Solving for x in Case 2

Let's solve for x in both cases.

(x-3) = 0

x = 3

(x-4) = 0

x = 4

Conclusion

We have found the x-intercepts of the given polynomial function $$y=(x-5)\left(x^2-7x+12\right)$$. The x-intercepts are $$(3, 0)$$ and $$(4, 0)$$. Therefore, the correct answer is:

B. $$(3, 0)$$ and C. $$(4, 0)$$

Note that the x-intercept $$(5, 0)$$ is not a solution to the equation, as it is not a root of the polynomial function.

Final Answer

The final answer is B. $$(3, 0)$$ and C. $$(4, 0)$$.

Introduction

In our previous article, we discussed how to find the x-intercept of a polynomial function. In this article, we will answer some frequently asked questions (FAQs) about finding the x-intercept of a polynomial function.

Q: What is the x-intercept of a polynomial function?

A: The x-intercept of a polynomial function is the point where the graph of the function crosses the x-axis. At this point, the y-coordinate is always 0.

Q: How do I find the x-intercept of a polynomial function?

A: To find the x-intercept of a polynomial function, you need to set the function equal to 0 and solve for x. You can use the zero product property to solve for x.

Q: What is the zero product property?

A: The zero product property states that if the product of two or more factors is equal to 0, then at least one of the factors must be equal to 0.

Q: How do I use the zero product property to find the x-intercept of a polynomial function?

A: To use the zero product property, you need to set the polynomial function equal to 0 and factor it. Then, you can set each factor equal to 0 and solve for x.

Q: What if the polynomial function cannot be factored?

A: If the polynomial function cannot be factored, you can use the quadratic formula to solve for x. The quadratic formula is:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations of the form $$ax^2 + bx + c = 0$$. The quadratic formula is:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula to find the x-intercept of a polynomial function?

A: To use the quadratic formula, you need to identify the values of a, b, and c in the polynomial function. Then, you can plug these values into the quadratic formula and solve for x.

Q: What if the quadratic formula gives me two solutions?

A: If the quadratic formula gives you two solutions, then the polynomial function has two x-intercepts.

Q: Can I use the quadratic formula to find the x-intercept of a polynomial function with a degree greater than 2?

A: No, the quadratic formula can only be used to find the x-intercept of a polynomial function with a degree of 2. If the polynomial function has a degree greater than 2, you will need to use a different method to find the x-intercept.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about finding the x-intercept of a polynomial function. We have discussed how to use the zero product property and the quadratic formula to find the x-intercept of a polynomial function.

Final Answer

The final answer is that the x-intercept of a polynomial function can be found using the zero product property and the quadratic formula.