A zero of a functionis a value of x that makes the function equal to zero. Plus, when a problem asks which function has zeros at x 10 and x 2, it is essentially asking for a mathematical expression that vanishes precisely at those two input values. Understanding how to construct such a function involves concepts from algebra, especially the factor theorem and polynomial behavior. This article walks you through the reasoning step‑by‑step, illustrates several concrete examples, and answers common questions that arise when dealing with zeros at specific points.
Not the most exciting part, but easily the most useful.
Understanding Zeros and Their Significance
Zeros, also called roots or x‑intercepts, are points where the graph of a function crosses the x‑axis. That said, for a single‑variable function (f(x)), a zero occurs when (f(x)=0). In elementary algebra, zeros are tightly linked to the factors of a polynomial: if (x-a) is a factor of a polynomial, then (a) is a zero of that polynomial The details matter here..
Key takeaway: Every zero corresponds to a linear factor of the function’s expression.
The Factor Theorem in Action
The factor theorem states that for a polynomial (p(x)),
- (x-a) is a factor of (p(x)) iff (p(a)=0).
That's why, to guarantee zeros at (x=10) and (x=2), the polynomial must contain the factors ((x-10)) and ((x-2)). Multiplying these factors together yields a basic polynomial that satisfies the requirement:
[ p(x) = (x-10)(x-2) ]
Expanding this product gives:
[ p(x) = x^{2} - 12x + 20 ]
This quadratic polynomial has exactly the two zeros we need, and no others (unless additional factors are introduced).
Constructing Functions with Zeros at 10 and 2
While the simplest polynomial is the quadratic above, there are infinitely many functions that share the same zeros. The general form can be expressed as:
[ f(x) = k,(x-10)(x-2) ]
where k is any non‑zero constant. Changing the value of k stretches or compresses the graph vertically but does not affect the locations of the zeros.
Variations to Explore
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Higher‑degree polynomials – Multiply the basic factor pair by additional polynomial terms, e.g.,
[ f(x) = (x-10)(x-2)(x+1)^{3} ] This introduces a third root at (-1) while preserving the zeros at 10 and 2 And that's really what it comes down to.. -
Rational functions – Place the factor pair in the denominator to create a function that is undefined at the zeros, e.g.,
[ g(x) = \frac{1}{(x-10)(x-2)} ]
Here the poles occur at the same x‑values, illustrating the reciprocal relationship And it works.. -
Piecewise definitions – Define a function that equals zero at 10 and 2 only on certain intervals, such as
[ h(x)=\begin{cases} (x-10)(x-2) & \text{if } x\ge 0\ 0 & \text{if } x<0 \end{cases} ]
This demonstrates that zeros can be controlled through domain restrictions.
Graphical Interpretation When plotted, any function that has zeros at (x=10) and (x=2) will intersect the x‑axis at those points. The shape of the curve near each zero depends on the multiplicity of the corresponding factor:
- Simple zero (multiplicity 1): The graph crosses the axis, changing sign.
- Even multiplicity: The graph merely touches the axis and rebounds, staying on the same side.
For the basic quadratic (p(x)=x^{2}-12x+20), both zeros are simple, so the parabola cuts the x‑axis at (x=2) and (x=10) and opens upward because the leading coefficient is positive Easy to understand, harder to ignore..
Practical Applications
Knowing how to engineer a function with prescribed zeros is valuable in several fields:
- Physics – Modeling phenomena where certain conditions must be met simultaneously, such as resonance frequencies.
- Engineering – Designing control systems where poles and zeros dictate system stability.
- Data fitting – Constructing interpolating polynomials that pass through given points, including zeros, to approximate complex datasets.
In each case, the ability to embed specific zeros ensures that the model behaves correctly under designated scenarios Not complicated — just consistent. That alone is useful..
Frequently Asked Questions
Q1: Must the function be a polynomial?
No. While polynomials provide a straightforward way to guarantee zeros, any function that equals zero at the desired inputs works. Examples include trigonometric functions like (\sin\big(\frac{\pi (x-2)}{8}\big)\sin\big(\frac{\pi (x-10)}{8}\big)), which also vanish at (x=2) and (x=10).
Q2: Can a function have more than two zeros at the same x‑value?
Yes, if a factor appears with a higher exponent. Here's a good example: ((x-10)^{2}(x-2)) yields a double zero at (x=10) and a simple zero at (x=2). The multiplicity influences how the graph behaves near that zero.
Q3: How does the leading coefficient affect the zeros?
The leading coefficient does not change the location of zeros; it only scales the function vertically. Still, it determines the end behavior of the polynomial (whether it rises to (+\infty) or falls to (-\infty) as (x) grows large).
Q4: What if I need a function that is zero at 10 and 2 but non‑zero elsewhere? A simple quadratic like ( (x-10)(x-2) ) satisfies this requirement. Any additional multiplicative factor that never equals zero (e.g., (e^{x})) will preserve the zeros while ensuring the function remains non‑zero elsewhere Simple as that..
Conclusion
Finding a function that has zeros at (x=10) and (x=2) is a direct application of the factor theorem. The minimal solution is the quadratic (x^{2}-12x+20), but the concept extends to infinitely many functions through multiplication by arbitrary non‑zero constants or additional polynomial terms. In real terms, by grasping how factors correspond to zeros, you can construct tailored functions for mathematical problems, scientific modeling, and engineering designs. This foundational skill not only answers the specific query which function has zeros at x 10 and x 2 but also equips you with a versatile tool for broader algebraic manipulations And that's really what it comes down to..
Conclusion
Finding a function that has zeros at (x=10) and (x=2) is a direct application of the factor theorem. The ability to intentionally create functions with desired zeros is a cornerstone of mathematical flexibility, enabling precise modeling and control in diverse fields. Also, this foundational skill not only answers the specific query which function has zeros at x 10 and x 2 but also equips you with a versatile tool for broader algebraic manipulations. So by grasping how factors correspond to zeros, you can construct tailored functions for mathematical problems, scientific modeling, and engineering designs. Because of that, the minimal solution is the quadratic (x^{2}-12x+20), but the concept extends to infinitely many functions through multiplication by arbitrary non‑zero constants or additional polynomial terms. As you continue to explore the world of functions and their properties, understanding this fundamental principle will get to a deeper appreciation for the power and elegance of mathematical expression The details matter here. No workaround needed..
Beyond that, the behavior near these points is dictated by the multiplicity of the factors. A zero with an odd multiplicity, such as the simple roots at 10 and 2 in our example, implies the graph crosses the x-axis. Conversely, an even multiplicity, like the squared term in ((x-10)^2(x-2)), causes the graph to touch the axis and turn back, creating a local extremum at that zero. This visual interpretation is crucial for sketching accurate graphs and understanding dynamic systems Easy to understand, harder to ignore..
Q3: How does the leading coefficient affect the zeros?
While the leading coefficient does not alter the location of the zeros, it plays a critical role in determining the polynomial's end behavior. A positive leading coefficient in an even-degree polynomial ensures the graph rises to positive infinity on both ends, whereas a negative coefficient flips this orientation. For odd-degree polynomials, the sign of the leading coefficient dictates whether the graph falls to the left and rises to the right, or the opposite.
Q4: What if I need a function that is zero at 10 and 2 but non‑zero elsewhere? A simple quadratic like ( (x-10)(x-2) ) satisfies this requirement. Any additional multiplicative factor that never equals zero (e.g., (e^{x})) will preserve the zeros while ensuring the function remains non‑zero elsewhere Not complicated — just consistent. Still holds up..
Conclusion
Finding a function that has zeros at (x=10) and (x=2) is a direct application of the factor theorem. By grasping how factors correspond to zeros, you can construct tailored functions for mathematical problems, scientific modeling, and engineering designs. Now, the ability to intentionally create functions with desired zeros is a cornerstone of mathematical flexibility, enabling precise modeling and control in diverse fields. The minimal solution is the quadratic (x^{2}-12x+20), but the concept extends to infinitely many functions through multiplication by arbitrary non‑zero constants or additional polynomial terms. Also, this foundational skill not only answers the specific query which function has zeros at x 10 and x 2 but also equips you with a versatile tool for broader algebraic manipulations. As you continue to explore the world of functions and their properties, understanding this fundamental principle will tap into a deeper appreciation for the power and elegance of mathematical expression.
