Which Function Represents A Function With Zeros At − 3 , − 1 , 0 -3, -1, 0 − 3 , − 1 , 0 , And 6 6 6 ?A. Y = ( X − 6 ) ( X + 1 ) ( X + 3 Y = (x-6)(x+1)(x+3 Y = ( X − 6 ) ( X + 1 ) ( X + 3 ] B. Y = X ( X − 3 ) ( X − 1 ) ( X + 6 Y = X(x-3)(x-1)(x+6 Y = X ( X − 3 ) ( X − 1 ) ( X + 6 ] C. Y = X ( X − 6 ) ( X + 1 ) ( X + 3 Y = X(x-6)(x+1)(x+3 Y = X ( X − 6 ) ( X + 1 ) ( X + 3 ] D. Y = ( X − 3 ) ( X − 1 ) ( X + 6 Y = (x-3)(x-1)(x+6 Y = ( X − 3 ) ( X − 1 ) ( X + 6 ]

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Understanding the Problem

When dealing with polynomial functions, the zeros of a function are the values of x that make the function equal to zero. In this problem, we are given four zeros: $-3, -1, 0$, and $6$. We need to determine which function represents a function with these zeros.

Factoring Polynomials

To find the function that represents a function with zeros at $-3, -1, 0$, and $6$, we can use the fact that if a polynomial has a zero at $x = a$, then $(x - a)$ is a factor of the polynomial. Therefore, if a polynomial has zeros at $-3, -1, 0$, and $6$, then it must have factors of $(x + 3), (x + 1), x$, and $(x - 6)$.

Analyzing the Options

Let's analyze each option to see which one has the correct factors.

Option A: $y = (x-6)(x+1)(x+3)$

This option has factors of $(x - 6), (x + 1)$, and $(x + 3)$. However, it is missing the factor $x$. Therefore, this option is not correct.

Option B: $y = x(x-3)(x-1)(x+6)$

This option has factors of $x, (x - 3), (x - 1)$, and $(x + 6)$. However, it is missing the factor $(x + 3)$. Therefore, this option is not correct.

Option C: $y = x(x-6)(x+1)(x+3)$

This option has factors of $x, (x - 6), (x + 1)$, and $(x + 3)$. However, it is missing the factor $(x - 1)$. Therefore, this option is not correct.

Option D: $y = (x-3)(x-1)(x+6)$

This option has factors of $(x - 3), (x - 1)$, and $(x + 6)$. However, it is missing the factors $x$ and $(x + 3)$. Therefore, this option is not correct.

Conclusion

None of the options A, B, C, or D have the correct factors. However, we can try to combine the factors from each option to see if we can get the correct function.

Combining Factors

Let's combine the factors from each option to see if we can get the correct function.

• Option A has factors of $(x - 6), (x + 1)$, and $(x + 3)$.
• Option B has factors of $(x - 3), (x - 1)$, and $(x + 6)$.
• Option C has factors of $x, (x - 6), (x + 1)$, and $(x + 3)$.
• Option D has factors of $(x - 3), (x - 1)$, and $(x + 6)$.

We can combine the factors from each option to get the following functions:

• $y = (x - 6)(x + 1)(x + 3)(x - 3)(x - 1)(x + 6)$
• $y = x(x - 6)(x + 1)(x + 3)(x - 3)(x - 1)(x + 6)$
• $y = x(x - 6)(x + 1)(x + 3)(x - 3)(x - 1)(x + 6)$
• $y = (x - 3)(x - 1)(x + 6)(x - 6)(x + 1)(x + 3)$

However, none of these functions have the correct factors.

The Correct Function

After analyzing each option and combining the factors, we can see that none of the options A, B, C, or D have the correct factors. However, we can try to find the correct function by using the fact that if a polynomial has zeros at $-3, -1, 0$, and $6$, then it must have factors of $(x + 3), (x + 1), x$, and $(x - 6)$.

The correct function is:

$y = (x + 3)(x + 1)x(x - 6)$

This function has the correct factors and represents a function with zeros at $-3, -1, 0$, and $6$.

Conclusion

In conclusion, the correct function that represents a function with zeros at $-3, -1, 0$, and $6$ is:

$y = (x + 3)(x + 1)x(x - 6)$

This function has the correct factors and represents a function with zeros at $-3, -1, 0$, and $6$.

Introduction

In our previous article, we discussed how to determine which function represents a function with zeros at $-3, -1, 0$, and $6$. We also found the correct function to be $y = (x + 3)(x + 1)x(x - 6)$. In this article, we will answer some frequently asked questions related to functions with zeros.

Q: What are the zeros of a function?

A: The zeros of a function are the values of x that make the function equal to zero. In other words, if a function has a zero at $x = a$, then $(x - a)$ is a factor of the function.

Q: How do I find the zeros of a function?

A: To find the zeros of a function, you can use the fact that if a polynomial has a zero at $x = a$, then $(x - a)$ is a factor of the polynomial. You can also use the Rational Root Theorem to find possible rational zeros of a polynomial.

Q: What is the Rational Root Theorem?

A: The Rational Root Theorem states that if a polynomial has a rational zero, then that rational zero must be a factor of the constant term divided by a factor of the leading coefficient.

Q: How do I use the Rational Root Theorem to find possible rational zeros?

A: To use the Rational Root Theorem, you need to find the factors of the constant term and the leading coefficient. Then, you can divide the factors of the constant term by the factors of the leading coefficient to find possible rational zeros.

Q: What is the difference between a zero and a root of a function?

A: A zero of a function is a value of x that makes the function equal to zero, while a root of a function is a value of x that makes the function equal to zero and is also a solution to the equation.

Q: How do I find the roots of a function?

A: To find the roots of a function, you can use the fact that if a polynomial has a root at $x = a$, then $(x - a)$ is a factor of the polynomial. You can also use the Rational Root Theorem to find possible rational roots of a polynomial.

Q: What is the relationship between zeros and roots of a function?

A: The zeros of a function are the values of x that make the function equal to zero, while the roots of a function are the values of x that make the function equal to zero and are also solutions to the equation. In other words, the zeros of a function are a subset of the roots of the function.

Q: How do I determine which function represents a function with zeros at $-3, -1, 0$, and $6$?

A: To determine which function represents a function with zeros at $-3, -1, 0$, and $6$, you can use the fact that if a polynomial has zeros at $-3, -1, 0$, and $6$, then it must have factors of $(x + 3), (x + 1), x$, and $(x - 6)$. You can then combine these factors to find the correct function.

Q: What is the correct function that represents a function with zeros at $-3, -1, 0$, and $6$?

A: The correct function that represents a function with zeros at $-3, -1, 0$, and $6$ is:

$y = (x + 3)(x + 1)x(x - 6)$

This function has the correct factors and represents a function with zeros at $-3, -1, 0$, and $6$.

Conclusion

In conclusion, we have answered some frequently asked questions related to functions with zeros. We have discussed how to find the zeros of a function, how to use the Rational Root Theorem to find possible rational zeros, and how to determine which function represents a function with zeros at $-3, -1, 0$, and $6$. We have also found the correct function to be $y = (x + 3)(x + 1)x(x - 6)$.