Select The Correct Answer.Which Point Is Where The Graph Of $y=(x-5)\left(x^2-7x+12\right)$ Crosses The $ X X X − A X I S -axis − A X I S ?A. $(-5,0)$ B. $(-3,0)$ C. $ ( 4 , 0 ) (4,0) ( 4 , 0 ) [/tex] D. $(12,0)$

by ADMIN 224 views

===========================================================

Understanding the Problem

The given problem involves finding the point where the graph of the polynomial function $y=(x-5)\left(x^2-7x+12\right)$ crosses the x-axis. This point is also known as the x-intercept of the function.

What is an x-Intercept?

An x-intercept is the point on the graph of a function where the function crosses the x-axis. At this point, the value of y is always 0.

How to Find the x-Intercept

To find the x-intercept of a polynomial function, we need to set the function equal to 0 and solve for x. In this case, we have the function $y=(x-5)\left(x^2-7x+12\right)$. We can start by setting y equal to 0 and then solving for x.

Solving the Equation

Setting y equal to 0, we get:

0=(x−5)(x2−7x+12)0=(x-5)\left(x^2-7x+12\right)

To solve for x, we can use the zero product property, which 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.

Factoring the Quadratic Expression

The quadratic expression $x^2-7x+12$ can be factored as:

(x−3)(x−4)(x-3)(x-4)

So, the equation becomes:

0=(x−5)((x−3)(x−4))0=(x-5)\left((x-3)(x-4)\right)

Solving for x

Using the zero product property, we can set each factor equal to 0 and solve for x:

x−5=0⇒x=5x-5=0 \Rightarrow x=5

(x−3)(x−4)=0⇒x−3=0 or x−4=0(x-3)(x-4)=0 \Rightarrow x-3=0 \text{ or } x-4=0

x−3=0⇒x=3x-3=0 \Rightarrow x=3

x−4=0⇒x=4x-4=0 \Rightarrow x=4

Finding the x-Intercept

Now that we have found the values of x, we can find the corresponding points on the graph. Since the value of y is always 0 at the x-intercept, we can write the points as:

(3,0),(4,0),(5,0)(3,0), (4,0), (5,0)

Conclusion

In conclusion, the graph of the polynomial function $y=(x-5)\left(x^2-7x+12\right)$ crosses the x-axis at the points $(3,0), (4,0), \text{ and } (5,0)$. Therefore, the correct answer is:

  • C. (4,0)

Note: The other options are not correct because the graph does not cross the x-axis at the points (−5,0),(−3,0), or (12,0)(-5,0), (-3,0), \text{ or } (12,0).

Final Answer

The final answer is C. (4,0).

=====================================================================================

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

A: The x-intercept of a polynomial function is the point on the graph where the function crosses the x-axis. At this point, the value of y 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 apply the zero product property to find the x-intercept?

A: To apply the zero product property, you need to set each factor equal to 0 and solve for x. This will give you the values of x where the function crosses the x-axis.

Q: What if the polynomial function has multiple factors?

A: If the polynomial function has multiple factors, you need to set each factor equal to 0 and solve for x. This will give you multiple values of x where the function crosses the x-axis.

Q: Can I use the quadratic formula to find the x-intercept?

A: Yes, you can use the quadratic formula to find the x-intercept of a quadratic function. However, if the function is a polynomial function with multiple factors, you may need to use the zero product property to find the x-intercept.

Q: How do I know which factor to set equal to 0 first?

A: You can set any factor equal to 0 first. The order in which you set the factors equal to 0 does not affect the final answer.

Q: Can I use a graphing calculator to find the x-intercept?

A: Yes, you can use a graphing calculator to find the x-intercept of a polynomial function. However, it's always a good idea to verify the answer by solving the equation algebraically.

Q: What if the polynomial function has no real solutions?

A: If the polynomial function has no real solutions, then the function does not cross the x-axis. In this case, the x-intercept is undefined.

Q: Can I find the x-intercept of a polynomial function with complex solutions?

A: Yes, you can find the x-intercept of a polynomial function with complex solutions. However, the x-intercept will be a complex number.

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

A: To find the x-intercept of a polynomial function with multiple x-intercepts, you need to set each factor equal to 0 and solve for x. This will give you multiple values of x where the function crosses the x-axis.

Q: Can I use the rational root theorem to find the x-intercept?

A: Yes, you can use the rational root theorem to find the x-intercept of a polynomial function. However, this method is only useful if you know the possible rational roots of the function.

Q: How do I know if a polynomial function has an x-intercept?

A: A polynomial function has an x-intercept if the function crosses the x-axis at some point. You can use the zero product property to determine if the function has an x-intercept.

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

A: Yes, you can find the x-intercept of a polynomial function with a degree greater than 2. However, the method you use will depend on the specific function and the number of x-intercepts.

Q: How do I find the x-intercept of a polynomial function with a leading coefficient of 1?

A: To find the x-intercept of a polynomial function with a leading coefficient of 1, you can use the zero product property to set each factor equal to 0 and solve for x.

Q: Can I use the factoring method to find the x-intercept?

A: Yes, you can use the factoring method to find the x-intercept of a polynomial function. However, this method is only useful if the function can be factored easily.

Q: How do I know if a polynomial function can be factored easily?

A: A polynomial function can be factored easily if it has a simple factorization, such as a difference of squares or a sum of cubes.

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

A: Yes, you can use the quadratic formula to find the x-intercept of a quadratic function. The quadratic formula is:

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Q: How do I apply the quadratic formula to find the x-intercept?

A: To apply the quadratic formula, you need to plug in the values of a, b, and c into the formula and simplify. This will give you the values of x where the function crosses the x-axis.

Q: Can I use the quadratic formula to find the x-intercept of a polynomial function with multiple factors?

A: No, you cannot use the quadratic formula to find the x-intercept of a polynomial function with multiple factors. The quadratic formula is only useful for quadratic functions.

Q: How do I know if a polynomial function has a rational x-intercept?

A: A polynomial function has a rational x-intercept if the x-intercept is a rational number. You can use the rational root theorem to determine if the x-intercept is rational.

Q: Can I use the rational root theorem to find the x-intercept of a polynomial function with multiple factors?

A: Yes, you can use the rational root theorem to find the x-intercept of a polynomial function with multiple factors. However, this method is only useful if you know the possible rational roots of the function.

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

A: To find the x-intercept of a polynomial function with a complex x-intercept, you need to use the quadratic formula or the factoring method. This will give you the complex values of x where the function crosses the x-axis.

Q: Can I use the factoring method to find the x-intercept of a polynomial function with a complex x-intercept?

A: Yes, you can use the factoring method to find the x-intercept of a polynomial function with a complex x-intercept. However, this method is only useful if the function can be factored easily.

Q: How do I know if a polynomial function has a complex x-intercept?

A: A polynomial function has a complex x-intercept if the x-intercept is a complex number. You can use the quadratic formula or the factoring method to determine if the x-intercept is complex.

Q: Can I use the quadratic formula to find the x-intercept of a polynomial function with a complex x-intercept?

A: Yes, you can use the quadratic formula to find the x-intercept of a polynomial function with a complex x-intercept. The quadratic formula is:

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

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

A: To apply the quadratic formula, you need to plug in the values of a, b, and c into the formula and simplify. This will give you the complex values of x where the function crosses the x-axis.

Q: Can I use the factoring method to find the x-intercept of a polynomial function with a complex x-intercept?

A: Yes, you can use the factoring method to find the x-intercept of a polynomial function with a complex x-intercept. However, this method is only useful if the function can be factored easily.

Q: How do I know if a polynomial function has a rational x-intercept and a complex x-intercept?

A: A polynomial function has a rational x-intercept and a complex x-intercept if the x-intercept is a rational number and a complex number. You can use the rational root theorem and the quadratic formula to determine if the x-intercept is rational and complex.

Q: Can I use the rational root theorem to find the x-intercept of a polynomial function with a rational x-intercept and a complex x-intercept?

A: Yes, you can use the rational root theorem to find the x-intercept of a polynomial function with a rational x-intercept and a complex x-intercept. However, this method is only useful if you know the possible rational roots of the function.

Q: How do I find the x-intercept of a polynomial function with a rational x-intercept and a complex x-intercept?

A: To find the x-intercept of a polynomial function with a rational x-intercept and a complex x-intercept,