Select The Correct Answer.Which Statement Best Describes The Zeros Of The Function $h(x)=(x-4)^2\left(x^2-7x+10\right$\]?A. The Function Has Four Complex Zeros. B. The Function Has Three Distinct Real Zeros. C. The Function Has Two Distinct

Introduction

In mathematics, a polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The zeros 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 zeros of a polynomial function and determine the correct statement that describes the zeros of the given function $h(x)=(x-4)^2\left(x^2-7x+10\right)$.

What are Zeros of a Polynomial Function?

The zeros of a polynomial function are the values of the variable that make the function equal to zero. In other words, if we substitute a zero of the function into the function, the result will be zero. For example, if we have a function $f(x)=x^2-4$, the zeros of the function are $x=2$ and $x=-2$, because when we substitute these values into the function, we get $f(2)=2^2-4=0$ and $f(-2)=(-2)^2-4=0$.

Types of Zeros

There are two types of zeros of a polynomial function: real zeros and complex zeros. Real zeros are the values of the variable that make the function equal to zero, and they can be either rational or irrational. Complex zeros, on the other hand, are the values of the variable that make the function equal to zero, but they are not real numbers. Complex zeros always come in conjugate pairs, which means that if $a+bi$ is a complex zero of a function, then $a-bi$ is also a complex zero of the function.

The Given Function

The given function is $h(x)=(x-4)^2\left(x^2-7x+10\right)$. To find the zeros of this function, we need to find the values of $x$ that make the function equal to zero. We can do this by setting the function equal to zero and solving for $x$.

Solving for Zeros

To solve for the zeros of the function, we can start by factoring the quadratic expression $x^2-7x+10$. We can factor this expression as $(x-5)(x-2)$. Therefore, the function can be written as $h(x)=(x-4)^2(x-5)(x-2)$.

Now, we can see that the function has three distinct factors: $(x-4)^2$, $(x-5)$, and $(x-2)$. Since the function is equal to zero when any of these factors is equal to zero, we can set each factor equal to zero and solve for $x$.

Finding the Zeros

Setting the first factor equal to zero, we get $(x-4)^2=0$. Solving for $x$, we get $x=4$. Setting the second factor equal to zero, we get $(x-5)=0$. Solving for $x$, we get $x=5$. Setting the third factor equal to zero, we get $(x-2)=0$. Solving for $x$, we get $x=2$.

Therefore, the zeros of the function are $x=4$, $x=5$, and $x=2$.

Conclusion

In conclusion, the zeros of the function $h(x)=(x-4)^2\left(x^2-7x+10\right)$ are $x=4$, $x=5$, and $x=2$. Since these zeros are distinct real numbers, the correct statement that describes the zeros of the function is:

• The function has three distinct real zeros.

This statement is correct because the function has three distinct real zeros, which are $x=4$, $x=5$, and $x=2$.

References

• [1] "Polynomial Functions." MathWorld, Wolfram Research, 2023.
• [2] "Zeros of a Polynomial Function." Math Open Reference, 2023.

Final Answer

The final answer is:

• B. The function has three distinct real zeros.
  Q&A: Understanding the Zeros of a Polynomial Function =====================================================

Introduction

In our previous article, we explored the concept of zeros of a polynomial function and determined the correct statement that describes the zeros of the given function $h(x)=(x-4)^2\left(x^2-7x+10\right)$. In this article, we will answer some frequently asked questions about the zeros of a polynomial function.

Q: What are the zeros of a polynomial function?

A: The zeros of a polynomial function are the values of the variable that make the function equal to zero. In other words, if we substitute a zero of the function into the function, the result will be zero.

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

A: To find the zeros of a polynomial function, we need to set the function equal to zero and solve for the variable. We can use various methods such as factoring, the quadratic formula, or synthetic division to find the zeros of the function.

Q: What are real zeros and complex zeros?

A: Real zeros are the values of the variable that make the function equal to zero, and they can be either rational or irrational. Complex zeros, on the other hand, are the values of the variable that make the function equal to zero, but they are not real numbers. Complex zeros always come in conjugate pairs.

Q: How do I determine if a zero is real or complex?

A: To determine if a zero is real or complex, we can use the discriminant of the quadratic equation. If the discriminant is positive, then the zero is real. If the discriminant is negative, then the zero is complex.

Q: Can a polynomial function have multiple zeros?

A: Yes, a polynomial function can have multiple zeros. In fact, a polynomial function can have any number of zeros, including zero, one, two, three, or more.

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

A: To find the multiplicity of a zero, we need to count the number of times the factor corresponding to the zero appears in the factored form of the polynomial function.

Q: What is the relationship between the zeros of a polynomial function and its graph?

A: The zeros of a polynomial function are the x-intercepts of its graph. In other words, if we graph the polynomial function, the zeros of the function will be the points where the graph intersects the x-axis.

Q: Can a polynomial function have zeros that are not real numbers?

A: Yes, a polynomial function can have zeros that are not real numbers. In fact, a polynomial function can have complex zeros, which are not real numbers.

Q: How do I determine if a polynomial function has real or complex zeros?

A: To determine if a polynomial function has real or complex zeros, we can use various methods such as factoring, the quadratic formula, or synthetic division. We can also use the discriminant of the quadratic equation to determine if a zero is real or complex.

Conclusion

In conclusion, the zeros of a polynomial function are the values of the variable that make the function equal to zero. We can find the zeros of a polynomial function by setting the function equal to zero and solving for the variable. We can also use various methods such as factoring, the quadratic formula, or synthetic division to find the zeros of the function. By understanding the zeros of a polynomial function, we can gain a deeper understanding of the behavior of the function and its graph.

References

• [1] "Polynomial Functions." MathWorld, Wolfram Research, 2023.
• [2] "Zeros of a Polynomial Function." Math Open Reference, 2023.

Final Answer

The final answer is:

• B. The function has three distinct real zeros.