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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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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.

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.

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 - 3*x* - 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 2*i*, so our solutions are *x* = 2*i* and *x* = -2*i*. 2*i* and -2*i* are imaginary numbers, so they are complex zeros.

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

- Convert the zeros to factors.
- Multiply the factors.
- 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.

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

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

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

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

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*.

Just in case, let's refresh your knowledge about imaginary zeros and conjugate pairs. Imaginary zeros of polynomial functions with real coefficients always occur in **conjugate pairs**. If (*a* + *bi*) (*a* and *b* are real numbers and *b* does not equal zero) is a zero of a polynomial function with real coefficients, then its conjugate (*a* - *bi*) is also a zero of the polynomial function.

Let's get back to the problem. First, let's change the zeros to factors. The zeros of 2, 2, *i*, and -*i* mean that our factors are as follows:

(*x* - 2), (*x* - 2), (*x* - *i*), and (*x* + *i*)

Next, we need to multiply the factors:

*f(x)* = (*x* - 2)(*x* - 2)(*x* - *i*)(*x* + *i*)

It might be best to multiply the first two factors first and then the last two factors separately.

(*x* - 2)(*x* - 2) = (*x*^2 - 4*x* + 4)

(*x* - *i*)(*x* + *i*) = (*x*^2 - *i*^2) *i*^2 = -1, so (*x*^2 - *i*^2) = (*x*^2 - (-1)) = (*x*^2 + 1)

We then can multiply these two products together.

*f(x)* = (*x*^2 - 4*x* + 4)(*x*^2 + 1)*f(x)* = *x*^4 - 4*x*^3 + 4*x*^2 + *x*^2 - 4*x* + 4

Finally, we combine like terms, which gives us:

*f(x)* = *x*^4 - 4*x*^3 + 5*x*^2 - 4*x* + 4

A polynomial function has real coefficients, a leading coefficient of 1, and the zeros 3 and (2 - *i*). Write a polynomial function of least degree in standard form. Remember that complex zeros occur in conjugate pairs; therefore, (2 + *i*) is also a zero.

First, let's change the zeros to factors. The zeros of 3, (2 - *i*), and (2 + *i*) mean that our factors are as follows:

(*x* - 3), (*x* - (2 - *i*)), and (*x* - (2 + *i*))

Next, we need to multiply the factors. It will be easier to multiply the complex factors first.

*f(x)* = (*x* - 3)(*x* - (2 - *i*))(*x* - (2 + *i*)) *f(x)* = (*x* - 3)(*x*^2 - 4*x* + 5) *f(x)* = *x*^3 - 4*x*^2 + 5*x* - 3*x*^2 + 12*x* - 15

Finally, we combine like terms:

*f(x)* = *x*^3 - 7*x*^2 + 17*x* - 15

In this lesson, you were given the process of converting a set of given zeros to a polynomial function. The steps were as follows:

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

This lesson considered polynomials with rational and/or complex zeros. Remember that complex zeros occur in conjugate pairs. A problem might only provide one of the complex zeros, leaving it up to the student to add its conjugate.

After you've reviewed this video lesson, you should be able to:

- List the steps to convert a set of zeros to a polynomial function
- Convert to a polynomial function given a set of rational and/or complex zeros

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

- How to Add, Subtract and Multiply Polynomials 6:53
- Factoring Polynomials Using Quadratic Form: Steps, Rules & Examples 8:38
- How to Divide Polynomials with Long Division 8:05
- How to Use Synthetic Division to Divide Polynomials 6:51
- Remainder Theorem & Factor Theorem: Definition & Examples 7:00
- Dividing Polynomials with Long and Synthetic Division: Practice Problems 10:11
- Finding Rational Zeros Using the Rational Zeros Theorem & Synthetic Division 8:45
- Fundamental Theorem of Algebra: Explanation and Example 7:39
- Using Rational & Complex Zeros to Write Polynomial Equations 8:59
- Go to Polynomial Functions of a Higher Degree

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