Find The Function With Real Zeros At X=3 And X=7
Hey guys! Let's dive into a fun math problem today. We're going to figure out which quadratic function has real zeros at x = 3 and x = 7. This means we're looking for a function that equals zero when we plug in these values for x. Understanding zeros of a function is crucial in algebra and calculus, as they represent the x-intercepts of the function's graph. This concept is not only fundamental in academic mathematics but also has practical applications in various fields such as physics, engineering, and economics, where finding the roots of equations can help in modeling real-world phenomena. For instance, engineers might use quadratic equations to model the trajectory of a projectile, while economists could use them to analyze cost and revenue curves. So, let's put on our thinking caps and explore how to solve this problem step by step!
Understanding Zeros and Factors
To start, it's super important to understand the connection between zeros and factors. When a function has a zero at x = a, it means that (x - a) is a factor of that function. Think of it like this: if plugging in x = a makes the function equal zero, then (x - a) must be part of the equation. This is a key concept in polynomial algebra and provides a direct link between the roots of a polynomial and its factored form. The factor theorem, a cornerstone of polynomial algebra, formalizes this relationship by stating that a polynomial f(x) has a factor (x - a) if and only if f(a) = 0. Understanding this connection allows us to move seamlessly between the roots of a polynomial and its algebraic expression, which is particularly useful in solving polynomial equations and sketching graphs of polynomial functions. Moreover, this understanding extends beyond simple quadratic functions and applies to polynomials of higher degrees, making it a versatile tool in mathematical analysis.
In our case, we have zeros at x = 3 and x = 7. So, the factors of our function must be (x - 3) and (x - 7). This immediately narrows down the possibilities and provides a concrete starting point for finding the correct function. By identifying these factors, we can construct a quadratic function that has the desired zeros and then compare it to the given options. This method highlights the power of factorization in solving algebraic problems and demonstrates how a deep understanding of fundamental concepts can simplify complex questions. Furthermore, this approach is not limited to finding quadratic functions; it can be applied to any polynomial function where the zeros are known.
Constructing the Function
Now that we know the factors, let's construct the function. We'll multiply the factors (x - 3) and (x - 7) together. This process is a straightforward application of the distributive property, which is a fundamental concept in algebra. The distributive property allows us to expand expressions by multiplying each term inside one set of parentheses by each term inside the other set, ensuring that we account for all possible combinations. This is a crucial skill in algebraic manipulation and is used extensively in solving equations, simplifying expressions, and working with polynomials. When we multiply factors, we are essentially reversing the process of factoring, which is another key technique in algebra. This back-and-forth movement between factored and expanded forms is essential for a comprehensive understanding of algebraic expressions.
So, let's do the math:
(x - 3)(x - 7) = x(x - 7) - 3(x - 7)
= x² - 7x - 3x + 21
= x² - 10x + 21
Boom! We've got our function. This function, f(x) = x² - 10x + 21, has factors (x - 3) and (x - 7), which means it will equal zero when x is 3 or 7. This confirms that our constructed function indeed has the desired zeros. The process of expanding the factors not only gives us the quadratic function but also reinforces the connection between the factored form and the standard form of a quadratic equation. This understanding is invaluable in various mathematical contexts, including solving quadratic equations, graphing quadratic functions, and analyzing their properties.
Evaluating the Given Options
Okay, we've built our function, f(x) = x² - 10x + 21. Now, let's evaluate the given options to see which one matches our result. This step is crucial to ensure that we have correctly identified the function with the specified zeros. Evaluating the options involves comparing our derived function with the provided choices, which may include functions with different coefficients or signs. This comparative analysis not only helps us confirm our answer but also reinforces our understanding of the relationship between the coefficients of a quadratic function and its roots.
The options given were:
- f(x) = x² + 4x - 21
- f(x) = x² - 4x - 21
- f(x) = x² - 10x + 21
- f(x) = x² - 10x - 21
By simply looking at these options, we can clearly see that the third option, f(x) = x² - 10x + 21, matches the function we constructed. This direct comparison highlights the importance of careful algebraic manipulation and attention to detail. Even a small difference in the coefficients or signs can significantly alter the roots of a quadratic equation, emphasizing the need for precision in mathematical calculations. This step also serves as a good practice in recognizing patterns and making quick comparisons, which are valuable skills in mathematical problem-solving.
Factoring to Verify
Just to be extra sure, we can also factor the given options and see which one gives us the factors (x - 3) and (x - 7). This is like double-checking our work to make sure we haven't made any silly mistakes. Factoring is the reverse process of expanding, and it's a fundamental skill in algebra. Being able to factor quadratic expressions allows us to easily identify their roots, which in turn helps us understand the behavior of the corresponding quadratic function. This process involves finding two binomials that, when multiplied together, give us the original quadratic expression. Factoring is not only a useful technique for solving equations but also for simplifying expressions and identifying key features of functions.
Let's factor the correct option, f(x) = x² - 10x + 21:
We're looking for two numbers that multiply to 21 and add up to -10. Those numbers are -3 and -7.
So, x² - 10x + 21 = (x - 3)(x - 7)
Yep, it checks out! Factoring confirms that our function indeed has zeros at x = 3 and x = 7. This step reinforces the importance of verifying our solutions and provides an alternative method for solving the problem. By factoring the quadratic expression, we can directly see the factors that correspond to the roots of the equation, providing a clear and intuitive understanding of the relationship between the algebraic expression and its solutions. This dual approach of constructing the function and then verifying it through factoring enhances our confidence in the answer and deepens our understanding of the underlying concepts.
Conclusion
So, the function with real zeros at x = 3 and x = 7 is f(x) = x² - 10x + 21. We found this by understanding the relationship between zeros and factors, constructing the function by multiplying the factors, evaluating the given options, and verifying our answer by factoring. You guys nailed it! This problem demonstrates how a solid understanding of algebraic principles can lead to a clear and efficient solution. The ability to identify factors from roots, construct polynomials from their factors, and verify solutions through factoring are essential skills in algebra and beyond. Mastering these techniques not only helps in solving specific problems but also provides a strong foundation for more advanced mathematical concepts.
Remember, math is like building with LEGOs – each concept builds on the previous one. Keep practicing, and you'll become math whizzes in no time! By consistently engaging with mathematical problems and reinforcing fundamental concepts, you can develop a deeper understanding and appreciation for the subject. Math is not just about memorizing formulas; it's about developing problem-solving skills and logical thinking, which are valuable assets in any field. So, keep exploring, keep questioning, and keep building your mathematical foundation!
