Find The Cubic Function With The Given Zeros

Finding the Cubic Function with Given Zeros

Finding a cubic function given its zeros is a fundamental concept in algebra. This article will guide you through the process, explaining the underlying principles and providing examples to solidify your understanding. We'll cover how to construct a cubic function from its roots, including cases with real and complex zeros, and explore how to incorporate additional information, such as a point on the curve, to fully define the function.

Understanding Zeros and Factors

The zeros (or roots) of a function are the values of x where the function's value, f(x), equals zero. For a cubic function, there are always three zeros, although they may not all be distinct or real numbers. Each zero corresponds to a factor of the cubic function. If r is a zero of the cubic function f(x), then (x - r) is a factor.

Constructing the Cubic Function

The general form of a cubic function is:

f(x) = a(x - r₁)(x - r₂)(x - r₃)

where:

• a is a non-zero constant that scales the function vertically.
• r₁, r₂, and r₃ are the three zeros of the function.

Example 1: Three Distinct Real Zeros

Let's find the cubic function with zeros at x = 1, x = 2, and x = -3.

1. Write the factors: (x - 1), (x - 2), and (x + 3)
2. Construct the function: f(x) = a(x - 1)(x - 2)(x + 3)
3. Determine 'a': Without additional information, we can't determine the value of a. The simplest cubic function is obtained by setting a = 1: f(x) = (x - 1)(x - 2)(x + 3) This expands to f(x) = x³ - 7x + 6. Any other non-zero value for a would yield a similar cubic function, simply scaled vertically.

Example 2: Repeated Zeros

Suppose the zeros are x = 1 (with multiplicity 2) and x = -2. This means the zero at x = 1 is a repeated root.

1. Write the factors: (x - 1)² and (x + 2)
2. Construct the function: f(x) = a(x - 1)²(x + 2)
3. Determine 'a': Again, setting a = 1 gives us the simplest form: f(x) = (x - 1)²(x + 2) which expands to f(x) = x³ - 3x + 2.

Example 3: Complex Zeros

Complex zeros always come in conjugate pairs. Let's say the zeros are x = 2 and x = 1 ± 2i.

1. Write the factors: (x - 2), (x - (1 + 2i)), and (x - (1 - 2i)).
2. Construct the function: f(x) = a(x - 2)(x - (1 + 2i))(x - (1 - 2i))
3. Simplify: Expanding the complex factors requires careful use of the difference of squares and results in: f(x) = a(x - 2)(x² - 2x + 5). Setting a = 1 provides a possible solution: f(x) = (x-2)(x² - 2x + 5) which expands to f(x) = x³ -4x² + 9x -10

Determining 'a' with an Additional Point

If you're given a point (x, y) that lies on the cubic function, you can use this information to solve for a. Substitute the x and y coordinates into the equation, and solve for a.

Conclusion

Finding a cubic function from its given zeros is a straightforward process involving factoring and expansion. Understanding the concept of multiplicity in repeated roots and handling complex zeros are crucial aspects of mastering this skill. Remember that the constant a can be determined by adding a point or leaving it as a variable for representing a family of functions with the same zeros. This knowledge empowers you to solve a wider range of algebraic problems and delve into more advanced mathematical concepts.