Which Function Has Zeros of 0 and 2?
Understanding which mathematical function possesses zeros at specific x‑values is a foundational skill in algebra and calculus. When a function’s graph crosses the x‑axis at a particular point, that point is called a zero (or root) of the function. In this article we will explore how to identify and construct functions that have the zeros 0 and 2, examine the relationship between zeros and factors, and provide concrete examples ranging from simple linear expressions to more complex polynomials. By the end, you will be able to determine the appropriate function form, manipulate its coefficients, and verify that the desired zeros are indeed present.
And yeah — that's actually more nuanced than it sounds.
Understanding Zeros of a Function A zero of a function (f(x)) is any value of (x) that makes the output (f(x)=0). Graphically, these are the x‑intercepts of the function’s curve. Algebraically, finding zeros involves solving the equation
[ f(x)=0 ]
for (x). The collection of all such solutions is often referred to as the root set of the function Turns out it matters..
Key points:
- Zeros can be real (crossing the x‑axis) or complex (occurring in conjugate pairs for polynomials with real coefficients).
- The multiplicity of a zero indicates how many times that root appears in the factorization of the function. A zero of multiplicity 1 is a simple root; multiplicity 2 yields a tangent touch, and so on.
- For polynomial functions, the Fundamental Theorem of Algebra guarantees that a degree‑(n) polynomial has exactly (n) complex roots (counting multiplicities).
How Zeros Relate to Factors There is a direct correspondence between the zeros of a polynomial and its factorized form. If (r) is a zero of a polynomial (p(x)), then ((x-r)) is a factor of (p(x)). So naturally, a polynomial that has zeros at (x = 0) and (x = 2) must contain the factors ((x-0) = x) and ((x-2)).
Thus, any polynomial that has 0 and 2 as zeros can be written as
[ p(x)=k \cdot x (x-2) \cdot q(x) ]
where (k) is a non‑zero constant (the leading coefficient) and (q(x)) is any polynomial that does not introduce additional zeros at the specified locations. The simplest case occurs when (q(x)=1), giving a quadratic function.
Constructing Functions with Given Zeros
The Basic Quadratic
The most straightforward function with zeros at 0 and 2 is a quadratic of the form
[ f(x)=a,x(x-2) ]
Expanding this expression yields
[ f(x)=a,(x^{2}-2x)=a x^{2}-2a x]
Here, (a) can be any non‑zero real number. Choosing (a=1) gives the canonical form
[\boxed{f(x)=x(x-2)=x^{2}-2x} ]
This function clearly satisfies (f(0)=0) and (f(2)=0). ### Adjusting the Leading Coefficient
Changing the value of (a) stretches or compresses the graph vertically but does not affect the location of the zeros. For example:
- If (a=3), then (f(x)=3x^{2}-6x).
- If (a=-\tfrac{1}{2}), then (f(x)=-\tfrac{1}{2}x^{2}+x).
All of these retain the same zeros at 0 and 2.
Adding Higher‑Degree Factors
If we wish to create a cubic, quartic, or higher‑degree polynomial that still has 0 and 2 as zeros, we can multiply the basic quadratic by any additional polynomial factor that does not introduce new zeros at those points. For instance:
[ g(x)=x(x-2)(x-5) ]
has zeros at 0, 2, and 5. To keep only 0 and 2, we could use a factor that evaluates to a non‑zero constant at those points, such as ((x^{2}+1)) or simply a constant (k).
[ h(x)=k,x(x-2)(x^{2}+1) ]
Here, the factor ((x^{2}+1)) never vanishes for real (x), so the only real zeros remain 0 and 2 Practical, not theoretical..
Example: Building a Quadratic with Zeros 0 and 2
Let’s walk through a concrete example step by step:
- Identify the required zeros: 0 and 2.
- Write the corresponding factors: (x) and ((x-2)).
- Multiply the factors: (x(x-2)).
- Choose a leading coefficient: Let’s pick (a=4) for a steeper parabola.
- Form the function:
[ f(x)=4x(x-2)=4x^{2}-8x ]
- Verify the zeros:
[ f(0)=4\cdot0^{2}-8\cdot0=0,\qquad f(2)=4\cdot2^{2}-8\cdot2=16-16=0 ]
Both checks confirm that the function indeed has the desired zeros.
Higher‑Degree Polynomials with Zeros 0 and 2
Beyond quadratics, we can embed the basic factor pair into more complex polynomials. Consider a cubic that must have 0 and 2 as zeros but also a third distinct zero at (x=7):
[ p(x)=k,x(x-2)(x-7) ]
If we set (k=1), the expanded form is [ p(x)=x^{3
Expanding the Cubic and Exploring Multiplicity
Continuing from the expression begun above, the full expanded form of the cubic [ p(x)=k,x(x-2)(x-7) ]
is obtained by first multiplying the two linear factors that we have already identified, then incorporating the third root:
[\begin{aligned} x(x-2) &= x^{2}-2x,\[4pt] (x^{2}-2x)(x-7) &= x^{3}-7x^{2}-2x^{2}+14x\ &= x^{3}-9x^{2}+14x . \end{aligned} ]
Thus, with (k=1),[ p(x)=x^{3}-9x^{2}+14x . ]
If we wish to preserve the zeros at (0) and (2) while allowing the third zero to vary, we simply replace the constant (7) with any parameter (c\neq 0,2). The general cubic then reads
[p_c(x)=k,x(x-2)(x-c)=k\bigl(x^{3}-(c+2)x^{2}+2c,x\bigr). ]
Changing the value of (c) slides the third intercept along the (x)-axis, whereas the coefficient (k) continues to control vertical scaling without moving any of the established zeros.
Multiplicity and Its Visual Impact
Sometimes a zero is required to be “touched” rather than crossed. This is achieved by assigning a multiplicity greater than one to a factor. To give you an idea, the factor (x^{2}) yields a double zero at the origin:
[ m(x)=k,x^{2}(x-2)=k,(x^{3}-2x^{2}) . ]
Here, the graph approaches the (x)-axis at (x=0) and bounces back, while still crossing at (x=2). By adjusting the exponent of either factor we can obtain triple, quadruple, or higher‑order contacts, each altering the curvature near the root while preserving the set of distinct zeros.
General Form for Any Degree
For a polynomial of arbitrary degree (n) that must have precisely the zeros (0) and (2) (and possibly other complex or real zeros that do not coincide with these), the most compact representation is
[ P(x)=k,x^{a}(x-2)^{b},Q(x), ]
where
- (a,b\in\mathbb{N}) denote the multiplicities of the zeros at (0) and (2) respectively, * (k\neq 0) is a scalar that controls vertical stretch, and
- (Q(x)) is any polynomial that never vanishes at (x=0) or (x=2) (for real‑valued functions, a common choice is a product of irreducible quadratics with positive discriminant, such as (x^{2}+1)).
This formulation guarantees that the only guaranteed real roots are (0) and (2); any additional roots lie in the zero set of (Q(x)) and can be placed wherever desired, provided they avoid the two distinguished points.
Concrete Illustrations
-
Quartic with a double root at the origin
[ r(x)=2,x^{2}(x-2)(x+1)=2\bigl(x^{4}-3x^{3}+2x^{2}\bigr). ] Zeros: (0) (multiplicity 2), (2), and (-1). -
Quintic that keeps only the prescribed real zeros [ s(x)=5,x(x-2)(x^{2}+4)=5\bigl(x^{4}-2x^{3}+4x^{2}-8x\bigr). ] Here (x^{2}+4) contributes a pair of complex conjugate roots (\pm 2i), leaving the real zeros unchanged Worth keeping that in mind..
-
Parameter‑driven family with a movable third root
[ t_{c}(x)=x(x-2)(x-c)+1, ] where the added constant (1) ensures that the polynomial never actually acquires an extra real root at (c); instead, the graph is simply shifted upward, preserving the original zeros while altering the shape of the curve.
Why This Construction Matters
Understanding how to embed prescribed zeros into a polynomial provides a powerful tool in many areas of mathematics and its applications:
- Signal processing – designing filters whose frequency response is governed by specific pole‑zero placements.
- Control theory – shaping characteristic equations so that system dynamics have desired damping characteristics.
