Applying the Factor Theorem to the Given Function

Oct 13, 2025 by talynt.org 90 views

Understanding the Basics of Function Zeros

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). The zeros of a function are the values of the input that result in an output of zero. In other words, they are the values of the input that make the function equal to zero.

The Factor Theorem: A Key Concept in Finding Zeros

The factor theorem is a fundamental concept in algebra that states that if a polynomial function f(x) has a factor (x - a), then f(a) = 0. This means that if we know that a function has a factor (x - a), we can conclude that a is a zero of the function.

Applying the Factor Theorem to the Given Function

In this problem, we are given that the function g has the factors (x - 7) and (x + 6). Using the factor theorem, we can conclude that g(7) = 0 and g(-6) = 0. This means that the zeros of the function g are x = 7 and x = -6.

Evaluating the Answer Choices

Now that we have found the zeros of the function g, let's evaluate the answer choices.

A. -7 and 6: This is not correct because we found that the zeros are x = 7 and x = -6, not x = -7 and x = 6.
B. -6 and 7: This is correct because we found that the zeros are x = 7 and x = -6, which matches this answer choice.
C. 6 and 7: This is not correct because we found that the zeros are x = 7 and x = -6, not x = 6 and x = 7.

Conclusion

In conclusion, the zeros of the function g are x = 7 and x = -6. This can be determined using the factor theorem, which states that if a polynomial function f(x) has a factor (x - a), then f(a) = 0. By applying this theorem to the given function, we can conclude that the zeros of the function g are x = 7 and x = -6.

Key Takeaways

The factor theorem states that if a polynomial function f(x) has a factor (x - a), then f(a) = 0.
The zeros of a function are the values of the input that result in an output of zero.
To find the zeros of a function, we can use the factor theorem and set each factor equal to zero.

Practice Problems

If function f has the factors (x - 3) and (x + 2), what are the zeros of function f?
If function h has the factors (x - 4) and (x + 5), what are the zeros of function h?
If function k has the factors (x - 1) and (x + 8), what are the zeros of function k?

Solutions

The zeros of function f are x = 3 and x = -2.
The zeros of function h are x = 4 and x = -5.
The zeros of function k are x = 1 and x = -8.
Frequently Asked Questions: Zeros of a Function =====================================================

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

A: In mathematics, a zero and a root of a function are often used interchangeably. However, some mathematicians make a distinction between the two terms. A zero of a function is a value of the input that results in an output of zero, whereas a root of a function is a value of the input that makes the function equal to zero. In other words, a zero is a specific value of the input, whereas a root is a solution to the equation.

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

A: To find the zeros of a function, you can use the factor theorem, which states that if a polynomial function f(x) has a factor (x - a), then f(a) = 0. You can also use other methods such as factoring, the quadratic formula, or numerical methods.

Q: What is the significance of finding the zeros of a function?

A: Finding the zeros of a function is important because it helps us understand the behavior of the function. The zeros of a function can tell us about the function's symmetry, periodicity, and other properties. Additionally, the zeros of a function can be used to solve equations and inequalities.

Q: Can a function have multiple zeros?

A: Yes, a function can have multiple zeros. For example, the function f(x) = (x - 2)(x - 3) has two zeros: x = 2 and x = 3.

Q: Can a function have no zeros?

A: Yes, a function can have no zeros. For example, the function f(x) = x^2 + 1 has no zeros because there is no value of x that makes the function equal to zero.

Q: How do I determine if a function has a zero?

A: To determine if a function has a zero, you can try to factor the function or use other methods such as the quadratic formula or numerical methods. You can also use the graphing calculator to visualize the function and see if it intersects the x-axis.

Q: Can a function have complex zeros?

A: Yes, a function can have complex zeros. For example, the function f(x) = x^2 + 1 has two complex zeros: x = i and x = -i.

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

A: To find the complex zeros of a function, you can use the quadratic formula or other methods such as factoring or numerical methods. You can also use the graphing calculator to visualize the function and see if it intersects the x-axis.

Q: Can a function have irrational zeros?

A: Yes, a function can have irrational zeros. For example, the function f(x) = x^2 - 2 has an irrational zero: x = √2.

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

A: To find the irrational zeros of a function, you can use the quadratic formula or other methods such as factoring or numerical methods. You can also use the graphing calculator to visualize the function and see if it intersects the x-axis.

Conclusion

In conclusion, finding the zeros of a function is an important concept in mathematics that can be used to solve equations and inequalities. By understanding the zeros of a function, we can gain insights into the behavior of the function and make predictions about its behavior.