Finding Zeros Of Functions How To Solve G(x) = (x^2 - 36)(x + 7)
In mathematics, a zero of a function is a value of the input variable that makes the function equal to zero. In simpler terms, it's the x-value where the graph of the function intersects the x-axis. Finding the zeros of a function is a fundamental concept in algebra and calculus, with applications in various fields, including physics, engineering, and economics. In this comprehensive guide, we'll walk through the process of finding the zeros of the function g(x) = (x^2 - 36)(x + 7), providing a step-by-step explanation and insights into the underlying principles.
Understanding the Function
Before diving into the solution, let's take a closer look at the function we're dealing with: g(x) = (x^2 - 36)(x + 7). This function is a polynomial function, specifically a cubic polynomial (degree 3), as it's a product of two factors: (x^2 - 36) and (x + 7). The factor (x^2 - 36) is a difference of squares, which can be further factored into (x - 6)(x + 6). This factorization will be crucial in finding the zeros of the function.
The key to finding the zeros of any function lies in setting the function equal to zero and solving for the variable (in this case, x). This is because the zeros are the values of x that make the function's output zero. When we have a function in factored form, like our g(x), the process becomes significantly easier. The zeros are the values of x that make each factor equal to zero. This is based on the zero-product property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. Understanding this property is essential for efficiently finding the zeros of polynomial functions.
The process of finding the zeros involves a few key steps. First, we set the function g(x) equal to zero. This gives us the equation (x^2 - 36)(x + 7) = 0. Next, we factor the expression, if possible. As mentioned earlier, (x^2 - 36) can be factored into (x - 6)(x + 6). So, our equation becomes (x - 6)(x + 6)(x + 7) = 0. Now, we apply the zero-product property. We set each factor equal to zero and solve for x. This will give us the zeros of the function. Finally, we can verify our solutions by plugging them back into the original function and confirming that the output is indeed zero.
Step-by-Step Solution
To find the zeros of the function g(x) = (x^2 - 36)(x + 7), we'll follow these steps:
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Set the function equal to zero: g(x) = 0 (x^2 - 36)(x + 7) = 0
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Factor the expression: The factor (x^2 - 36) is a difference of squares, which can be factored as (x - 6)(x + 6). (x - 6)(x + 6)(x + 7) = 0
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Apply the zero-product property: The zero-product property states that if the product of several factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero: x - 6 = 0 x + 6 = 0 x + 7 = 0
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Solve for x: Solving each equation for x, we get: x = 6 x = -6 x = -7
Therefore, the zeros of the function g(x) are 6, -6, and -7.
Analyzing the Options
Now that we've found the zeros of the function, let's analyze the given options to determine the correct answer:
A. 18
B. -6
C. 0
D. 7
Comparing our calculated zeros (6, -6, and -7) with the options, we can see that option B (-6) is one of the zeros of the function. The other zeros, 6 and -7, are not listed as options. To confirm, we can plug -6 back into the function:
g(-6) = ((-6)^2 - 36)(-6 + 7)
g(-6) = (36 - 36)(1)
g(-6) = (0)(1)
g(-6) = 0
This confirms that -6 is indeed a zero of the function.
Why Other Options Are Incorrect
Let's briefly explain why the other options are incorrect:
A. 18: Plugging 18 into the function does not result in zero. g(18) = ((18)^2 - 36)(18 + 7) ≠0
C. 0: Plugging 0 into the function does not result in zero. g(0) = ((0)^2 - 36)(0 + 7) = (-36)(7) ≠0
D. 7: Plugging 7 into the function does not result in zero. g(7) = ((7)^2 - 36)(7 + 7) = (13)(14) ≠0
Conclusion
In conclusion, the correct answer is B. -6. We found the zeros of the function g(x) = (x^2 - 36)(x + 7) by setting the function equal to zero, factoring the expression, applying the zero-product property, and solving for x. This process yielded the zeros 6, -6, and -7. By comparing these zeros with the given options, we identified -6 as the correct answer. Understanding how to find the zeros of a function is a crucial skill in mathematics, with applications in various fields. This step-by-step guide provides a clear and concise approach to solving such problems.
Remember, finding the zeros of a function is a fundamental concept that builds the foundation for more advanced topics in mathematics. By mastering this skill, you'll be well-equipped to tackle a wide range of problems in algebra, calculus, and beyond. Keep practicing, and you'll become proficient in finding the zeros of any function you encounter!
