Which Of The Following Describes The Zeroes Of The Graph Of F ( X ) = 3 X 6 + 30 X 5 + 75 X 4 F(x)=3 X^6+30 X^5+75 X^4 F ( X ) = 3 X 6 + 30 X 5 + 75 X 4 ?A. -5 With Multiplicity 2 And 1 3 \frac{1}{3} 3 1 ​ With Multiplicity 4B. 5 With Multiplicity 2 And 1 3 \frac{1}{3} 3 1 ​ With Multiplicity 4C. -5 With

In mathematics, a polynomial function is a function that can be expressed as a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. The zeroes of a polynomial function are the values of the variable that make the function equal to zero. In this article, we will explore the concept of zeroes of a polynomial function and determine which of the given options describes the zeroes of the graph of $f(x)=3 x^6+30 x^5+75 x^4$.

What are Zeroes of a Polynomial Function?

The zeroes of a polynomial function are the values of the variable that make the function equal to zero. In other words, if we have a polynomial function $f(x)$, then the zeroes of $f(x)$ are the values of $x$ such that $f(x) = 0$. For example, if we have a polynomial function $f(x) = x^2 - 4$, then the zeroes of $f(x)$ are $x = 2$ and $x = -2$, because $f(2) = 2^2 - 4 = 0$ and $f(-2) = (-2)^2 - 4 = 0$.

How to Find Zeroes of a Polynomial Function

There are several methods to find the zeroes of a polynomial function, including factoring, synthetic division, and the rational root theorem. Factoring involves expressing the polynomial as a product of simpler polynomials, while synthetic division involves dividing the polynomial by a linear factor. The rational root theorem states that if a rational number $p/q$ is a root of the polynomial, then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient.

The Given Polynomial Function

The given polynomial function is $f(x)=3 x^6+30 x^5+75 x^4$. To find the zeroes of this function, we can try to factor it or use synthetic division. However, factoring a sixth-degree polynomial can be challenging, so we will use the rational root theorem to find the possible rational roots.

Using the Rational Root Theorem

The rational root theorem states that if a rational number $p/q$ is a root of the polynomial, then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient. In this case, the constant term is 0 and the leading coefficient is 3. Therefore, the possible rational roots are $\pm 1, \pm 3, \pm 5, \pm 15, \pm 25, \pm 75$.

Finding the Zeroes

To find the zeroes of the polynomial function, we can try to divide it by each of the possible rational roots. We can use synthetic division or long division to divide the polynomial by each of the possible rational roots. After trying each of the possible rational roots, we find that $x = -5$ and $x = 1/3$ are the zeroes of the polynomial function.

Determining the Multiplicity of the Zeroes

The multiplicity of a zero is the number of times the zero is repeated in the factorization of the polynomial. In this case, we find that $x = -5$ is a zero with multiplicity 2 and $x = 1/3$ is a zero with multiplicity 4.

Conclusion

In conclusion, the zeroes of the graph of $f(x)=3 x^6+30 x^5+75 x^4$ are $x = -5$ with multiplicity 2 and $x = 1/3$ with multiplicity 4. Therefore, the correct answer is option A.

Final Answer

In our previous article, we explored the concept of zeroes of a polynomial function and determined which of the given options describes the zeroes of the graph of $f(x)=3 x^6+30 x^5+75 x^4$. In this article, we will answer some frequently asked questions related to the zeroes of a polynomial function.

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

A: A zero and a root of a polynomial function are the same thing. They refer to the values of the variable that make the function equal to zero.

Q: How do I find the zeroes of a polynomial function?

A: There are several methods to find the zeroes of a polynomial function, including factoring, synthetic division, and the rational root theorem. Factoring involves expressing the polynomial as a product of simpler polynomials, while synthetic division involves dividing the polynomial by a linear factor. The rational root theorem states that if a rational number $p/q$ is a root of the polynomial, then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient.

Q: What is the multiplicity of a zero?

A: The multiplicity of a zero is the number of times the zero is repeated in the factorization of the polynomial. For example, if a polynomial function has a zero with multiplicity 2, it means that the zero is repeated twice in the factorization of the polynomial.

Q: How do I determine the multiplicity of a zero?

A: To determine the multiplicity of a zero, you need to factor the polynomial and count the number of times the zero is repeated. You can also use the rational root theorem to find the possible rational roots and then use synthetic division or long division to divide the polynomial by each of the possible rational roots.

Q: Can a polynomial function have more than one zero with the same multiplicity?

A: Yes, a polynomial function can have more than one zero with the same multiplicity. For example, a polynomial function can have two zeros with multiplicity 2, or three zeros with multiplicity 4.

Q: How do I graph a polynomial function with multiple zeroes?

A: To graph a polynomial function with multiple zeroes, you need to find the zeroes of the function and then use the multiplicity of each zero to determine the shape of the graph. For example, if a polynomial function has a zero with multiplicity 2, the graph will have a double root at that point.

Q: Can a polynomial function have a zero with a negative multiplicity?

A: No, a polynomial function cannot have a zero with a negative multiplicity. The multiplicity of a zero is always a non-negative integer.

Q: How do I use the rational root theorem to find the zeroes of a polynomial function?

A: To use the rational root theorem to find the zeroes of a polynomial function, you need to find the possible rational roots by dividing the constant term by the leading coefficient. Then, you can use synthetic division or long division to divide the polynomial by each of the possible rational roots.

Conclusion

In conclusion, the zeroes of a polynomial function are the values of the variable that make the function equal to zero. The multiplicity of a zero is the number of times the zero is repeated in the factorization of the polynomial. By understanding the zeroes and multiplicity of a polynomial function, you can graph the function and determine its behavior.

Final Answer

The final answer is A.