Finding Zeros Of The Function F(x) = 5(x+2) / 3(x-1)(x+7)

Zeros of a function are the values of x that make the function equal to zero. In other words, they are the x-values where the graph of the function intersects the x-axis. Finding the zeros of a function is a fundamental concept in mathematics, especially in algebra and calculus. In this article, we will delve into the process of identifying the zeros of the given rational function:

$F(x)=\frac{5(x+2)}{3(x-1)(x+7)}$

This detailed explanation will guide you through the steps to correctly identify the zeros of this particular function, ensuring a solid understanding of the underlying principles.

Identifying Zeros of Rational Functions

To find the zeros of a rational function, we need to determine the values of x for which the function F(x) equals zero. A rational function is a function that can be expressed as the quotient of two polynomials. For a rational function to be zero, the numerator must be zero, while the denominator must not be zero. This is because any number divided by zero is undefined, and thus, a zero of a function cannot occur where the denominator is also zero. Understanding this principle is crucial for correctly identifying the zeros of any rational function. The function we are examining is:

$F(x)=\frac{5(x+2)}{3(x-1)(x+7)}$

Here, the numerator is $5(x+2)$, and the denominator is $3(x-1)(x+7)$. We will explore each part to find the values of x that make the function zero.

Setting the Numerator to Zero

The first step in finding the zeros of the given function is to set the numerator equal to zero. The numerator of our function is $5(x+2)$. So, we need to solve the equation:

$5(x+2) = 0$

To solve this, we first divide both sides by 5:

$x+2 = 0$

Then, we subtract 2 from both sides:

$x = -2$

So, one potential zero of the function is $x = -2$. This means that when x is -2, the numerator of the function becomes zero, which makes the entire function zero, provided the denominator is not also zero at this point. We will verify this in the next step by checking the denominator.

Checking the Denominator

After finding the potential zero from the numerator, it is crucial to check whether this value makes the denominator zero as well. The denominator of our function is $3(x-1)(x+7)$. If the denominator is zero for the same value of x that makes the numerator zero, then that value is not a zero of the function, but rather a point of discontinuity (a hole or a vertical asymptote). Let's substitute $x = -2$ into the denominator:

$3(-2-1)(-2+7) = 3(-3)(5) = -45$

Since the denominator is -45 when $x = -2$, it is not zero. Therefore, $x = -2$ is indeed a zero of the function. This confirms that when x is -2, the function F(x) equals zero, making it a valid zero of the function. This step is critical to ensure that the identified zeros are actual zeros and not points of discontinuity.

Identifying Other Potential Zeros

To ensure we have found all the zeros of the function, we need to consider the entire structure of the rational function. The function is:

$F(x)=\frac{5(x+2)}{3(x-1)(x+7)}$

We have already found that setting the factor $(x+2)$ in the numerator to zero gives us a zero of the function. Now, we must also consider the factors in the denominator, $(x-1)$ and $(x+7)$. These factors will help us identify any values of x that would make the denominator zero, which are not zeros of the function but are critical points to consider.

Values that Make the Denominator Zero

Let's determine the values of x that make the denominator $3(x-1)(x+7)$ equal to zero. This occurs when either $(x-1) = 0$ or $(x+7) = 0$. Solving these equations:

For $(x-1) = 0$:

$x = 1$

For $(x+7) = 0$:

$x = -7$

Thus, the denominator is zero when $x = 1$ or $x = -7$. These values are not zeros of the function because they make the function undefined (division by zero). Instead, they indicate vertical asymptotes, which are points where the function approaches infinity or negative infinity. Understanding these points is crucial for sketching the graph of the function and analyzing its behavior.

Conclusion

In conclusion, to find the zeros of the rational function:

$F(x)=\frac{5(x+2)}{3(x-1)(x+7)}$

we set the numerator equal to zero and solved for x:

$5(x+2) = 0$ which gives $x = -2$.

We then verified that this value does not make the denominator zero. The values that make the denominator zero, $x = 1$ and $x = -7$, are not zeros of the function but rather points of discontinuity.

Therefore, the only zero of the function is $x = -2$. Understanding how to find zeros and identify points of discontinuity is essential for analyzing and graphing rational functions. This process ensures a comprehensive understanding of the function's behavior across its domain.

The correct answer is B. -2.

The other options are incorrect:

• A. -7: This value makes the denominator zero, not the numerator.
• C. 1: This value also makes the denominator zero.
• D. -3: Substituting -3 into the function does not result in zero.
• E. 2: Substituting 2 into the function does not result in zero.
• F. 3: Substituting 3 into the function does not result in zero.

Therefore, only option B, -2, is the zero of the given function. This detailed explanation clarifies the steps and reasoning behind identifying the correct zero of the rational function.