Select The Correct Answer.What Are The X X X -intercepts Of F ( X ) = X 3 + 4 X 2 − 9 X − 36 F(x) = X^3 + 4x^2 - 9x - 36 F ( X ) = X 3 + 4 X 2 − 9 X − 36 ?A. ( − 6 , 0 ) , ( 2 , 0 ) , ( 3 , 0 (-6, 0), (2, 0), (3, 0 ( − 6 , 0 ) , ( 2 , 0 ) , ( 3 , 0 ] B. ( − 1 , 0 ) , ( 2 , 0 ) , ( 18 , 0 (-1, 0), (2, 0), (18, 0 ( − 1 , 0 ) , ( 2 , 0 ) , ( 18 , 0 ] C. ( − 4 , 0 ) , ( − 3 , 0 ) , ( 3 , 0 (-4, 0), (-3, 0), (3, 0 ( − 4 , 0 ) , ( − 3 , 0 ) , ( 3 , 0 ] D. $(1, 0), (2, 0), (3,

Understanding the Problem

The problem requires finding the $x$-intercepts of a given cubic function, $f(x) = x^3 + 4x^2 - 9x - 36$. The $x$-intercepts are the points where the graph of the function crosses the $x$-axis, and they can be found by solving the equation $f(x) = 0$.

What are $x$-Intercepts?

The $x$-intercepts of a function are the values of $x$ where the function's graph intersects the $x$-axis. In other words, they are the solutions to the equation $f(x) = 0$. The $x$-intercepts are also known as the roots or zeros of the function.

How to Find $x$-Intercepts

To find the $x$-intercepts of a function, we need to solve the equation $f(x) = 0$. This can be done using various methods, including factoring, the quadratic formula, and numerical methods. In this case, we will use factoring to find the $x$-intercepts of the given cubic function.

Factoring the Cubic Function

The given cubic function is $f(x) = x^3 + 4x^2 - 9x - 36$. To factor this function, we need to find three numbers whose product is $-36$ and whose sum is $4$. These numbers are $-6$, $-3$, and $1$, since $(-6) \cdot (-3) \cdot 1 = -36$ and $(-6) + (-3) + 1 = -8$, which is close to $4$. However, we can try to find a combination of three numbers that satisfy both conditions.

After some trial and error, we find that the correct combination of numbers is $-6$, $3$, and $-3$, since $(-6) \cdot 3 \cdot (-3) = -54$, which is not equal to $-36$. However, we can try to factor the function as $(x + 6)(x^2 - 3x - 6)$.

Factoring the Quadratic Function

The quadratic function $x^2 - 3x - 6$ can be factored as $(x - 6)(x + 1)$. Therefore, the factored form of the cubic function is $(x + 6)(x - 6)(x + 1)$.

Finding the $x$-Intercepts

To find the $x$-intercepts of the function, we need to set each factor equal to zero and solve for $x$. This gives us the following equations:

• $(x + 6) = 0 \Rightarrow x = -6$
• $(x - 6) = 0 \Rightarrow x = 6$
• $(x + 1) = 0 \Rightarrow x = -1$

Therefore, the $x$-intercepts of the function are $(-6, 0)$, $(6, 0)$, and $(-1, 0)$.

Conclusion

In conclusion, the $x$-intercepts of the given cubic function $f(x) = x^3 + 4x^2 - 9x - 36$ are $(-6, 0)$, $(6, 0)$, and $(-1, 0)$. These points represent the values of $x$ where the graph of the function crosses the $x$-axis.

Answer

Q: What are $x$-intercepts?

A: The $x$-intercepts of a function are the values of $x$ where the function's graph intersects the $x$-axis. In other words, they are the solutions to the equation $f(x) = 0$. The $x$-intercepts are also known as the roots or zeros of the function.

Q: How do I find the $x$-intercepts of a function?

A: To find the $x$-intercepts of a function, you need to solve the equation $f(x) = 0$. This can be done using various methods, including factoring, the quadratic formula, and numerical methods.

Q: What is factoring, and how do I use it to find $x$-intercepts?

A: Factoring is a method of solving equations by expressing them as a product of simpler expressions. To factor a function, you need to find the factors of the function's coefficients and the constant term. Once you have factored the function, you can set each factor equal to zero and solve for $x$ to find the $x$-intercepts.

Q: Can you give an example of factoring a cubic function?

A: Yes, let's consider the cubic function $f(x) = x^3 + 4x^2 - 9x - 36$. To factor this function, we need to find three numbers whose product is $-36$ and whose sum is $4$. After some trial and error, we find that the correct combination of numbers is $-6$, $3$, and $-3$, since $(-6) \cdot 3 \cdot (-3) = -54$, which is not equal to $-36$. However, we can try to factor the function as $(x + 6)(x^2 - 3x - 6)$.

Q: How do I factor the quadratic function $x^2 - 3x - 6$?

A: The quadratic function $x^2 - 3x - 6$ can be factored as $(x - 6)(x + 1)$. Therefore, the factored form of the cubic function is $(x + 6)(x - 6)(x + 1)$.

Q: How do I find the $x$-intercepts of the factored function?

A: To find the $x$-intercepts of the factored function, we need to set each factor equal to zero and solve for $x$. This gives us the following equations:

• $(x + 6) = 0 \Rightarrow x = -6$
• $(x - 6) = 0 \Rightarrow x = 6$
• $(x + 1) = 0 \Rightarrow x = -1$

Therefore, the $x$-intercepts of the function are $(-6, 0)$, $(6, 0)$, and $(-1, 0)$.

Q: What if I don't know how to factor a function?

A: If you don't know how to factor a function, you can use other methods to find the $x$-intercepts, such as the quadratic formula or numerical methods. Alternatively, you can try using a graphing calculator or online tool to find the $x$-intercepts.

Q: Can you give an example of using the quadratic formula to find $x$-intercepts?

A: Yes, let's consider the quadratic function $f(x) = x^2 + 4x + 4$. To find the $x$-intercepts of this function, we can use the quadratic formula:

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

In this case, $a = 1$, $b = 4$, and $c = 4$. Plugging these values into the formula, we get:

$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(4)}}{2(1)}$ $x = \frac{-4 \pm \sqrt{16 - 16}}{2}$ $x = \frac{-4 \pm \sqrt{0}}{2}$ $x = \frac{-4}{2}$ $x = -2$

Therefore, the $x$-intercept of the function is $(-2, 0)$.

Q: What if I have a rational function, and I want to find the $x$-intercepts?

A: If you have a rational function, and you want to find the $x$-intercepts, you can set the numerator of the function equal to zero and solve for $x$. This will give you the $x$-intercepts of the function.

Q: Can you give an example of finding the $x$-intercepts of a rational function?

A: Yes, let's consider the rational function $f(x) = \frac{x^2 - 4}{x + 2}$. To find the $x$-intercepts of this function, we can set the numerator equal to zero and solve for $x$:

$x^2 - 4 = 0$ $x^2 = 4$ $x = \pm 2$

Therefore, the $x$-intercepts of the function are $(-2, 0)$ and $(2, 0)$.

Conclusion

In conclusion, finding the $x$-intercepts of a function is an important concept in algebra and calculus. There are various methods to find the $x$-intercepts, including factoring, the quadratic formula, and numerical methods. By understanding these methods, you can solve a wide range of problems and applications in mathematics and science.