Using Rational & Complex Zeros to Write Polynomial Equations

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  • 0:01 Turning Zeros into Functions
  • 1:12 Definition
  • 2:44 Process Steps
  • 3:26 Examples
  • 8:14 Lesson Summary
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Lesson Transcript
Instructor: David Liano
In this lesson, you will learn how to write a polynomial function from its given zeros. You will learn how to follow a process that converts zeros into factors and then factors into polynomial functions.

Turning Zeros into Functions

A typical part of an algebra course is learning how to factor and find solutions of polynomial functions, such as quadratic equations. You often also learn how to factor and find solutions of polynomial functions of degree 3 or higher. However, in this lesson, you will be completing this process in reverse. After completing this lesson, you will be able to write a polynomial function from a given set of zeros.

Let's think of factoring as separating a water molecule into its basic parts of hydrogen and oxygen. If you remember what you learned from your chemistry class in high school, water is separated into two parts hydrogen and one part oxygen.

The reverse of this process is the combining of hydrogen gas and oxygen gas to form water. This is sort of what we will be doing when we write a polynomial function from a given set of zeros. Think of the hydrogen and oxygen molecules as the zeros and the water molecule as the polynomial function.

Definitions

We should recall that the zeros of a polynomial function are the numbers that solve the equation f(x) = 0. These numbers are also sometimes referred to as roots or solutions.

A rational zero is simply a solution that is a rational number, which is a number that can be written as a fraction of two integers. For instance, let's consider the quadratic function f(x) = x^2 - 3x - 28. In factored form, this function equals f(x) = (x + 4)(x - 7). The zero product property tells us that the solutions are x = -4 and x = 7. These are rational zeros because -4 and 7 are rational numbers.

A complex zero is a solution that has an imaginary part. For instance, let's consider the quadratic function f(x) = x^2 + 4.

If we make the function equal to zero and solve for x, we will end up with the square root of a negative number, which means that we will have complex zeros. The square root of -4 can be simplified to 2i, so our solutions are x = 2i and x = -2i. 2i and -2i are imaginary numbers, so they are complex zeros.

Process Steps

We will use the following steps to write a polynomial function from its given zeros:

  1. Convert the zeros to factors.
  2. Multiply the factors.
  3. Combine like terms and write with powers of x in descending order, which is the standard form of a polynomial function.

We will now complete some examples. Our examples will state that we will be writing polynomials of least degree that have real numbers for coefficients and a leading coefficient of 1. These stipulations are pretty typical for the problems we will be solving. These conditions ensure only one possible answer to our problems.

Example 1

A polynomial function has real coefficients, a leading coefficient of 1, and the zeros -1, -2, and 5. Write a polynomial function of least degree in standard form.

First, let's change the zeros to factors. The rational zeros of -1, -2, and 5 mean that our factors are as follows:

(x + 1), (x + 2), and (x - 5)

Next, we need to multiply the factors:

f(x) = (x + 1)(x + 2)(x - 5)
f(x) = (x^2 + 3x + 2)(x - 5)
f(x) = x^3 + 3x^2 + 2x - 5x^2 - 15x - 10

Our next step is to combine like terms, which gives us:

f(x) = x^3 - 2x^2 - 13x - 10

There we have it, a polynomial function in standard form.

Example 2

A polynomial function has real coefficients, a leading coefficient of 1, and the zeros 2, 2, i, and -i. Write a polynomial function of least degree in standard form.

You might have noticed that this function has a repeated zero, 2, and two imaginary zeros, i and -i.

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