Lesson Objectives
• Learn how to write a polynomial function given zeros and a point

## How to Write a Polynomial Equation Given Zeros and a Point

In this lesson, we will learn about the fundamental theorem of algebra and how to write a polynomial equation. The fundamental theorem of algebra tells us that a polynomial function of degree n will have n complex solutions. This means a 1st degree equation would have 1 solution, a 2nd degree equation will have 2 solutions, and so on and so forth. Now, these solutions may be repeats. In this case, you will see the term multiplicity used.
We can use these facts to write a polynomial equation given certain conditions.
Suppose our polynomial function has a degree of 3, and has the following zeros: $$2, 5, 1$$ Additionally, we are told that f(0) = -20.
Let's set up a polynomial function: $$f(x)=a(x - k_1)(x - k_2)(x - k_3)$$ Now, let's plug in for the zeros. The order does not matter: $$f(x)=a(x - 2)(x - 5)(x - 1)$$ $$f(x)=a(x^3 - 8x^2 + 17x - 10)$$ To find a, use the fact that f(0) is -20. $$-20=a(-10)$$ $$a=\frac{-20}{-10}=2$$ We know that a is 2: $$f(x)=2(x^3 - 8x^2 + 17x - 10)$$ $$f(x)=2x^3 - 16x^2 + 34x - 20$$

#### Skills Check:

Example #1

Write a polynomial function of degree 3. $$\text{Zeros}: 4, 2, -5$$ $$f(0)=40$$

A
$$f(x)=-7x^3 + 5x^2 - 8x + 1$$
B
$$f(x)=x^3 - 3x^2 + 9x + 1$$
C
$$f(x)=x^3 - x^2 - 22x + 40$$
D
$$f(x)=-2x^3 - 3x^2 - 11x + 6$$
E
$$f(x)=2x^3 - 3x^2 - 11x + 6$$

Example #2

Write a polynomial function of degree 3.

Please choose the best answer. $$\text{Zeros}: 3 \hspace{.2em}\text{mult.}\hspace{.2em}2, 1$$ $$f(0)=-9$$

A
$$f(x)=-5x^3 + 2x^2 + 3x - 4$$
B
$$f(x)=-x^3 + 5x^2 + 2x + 1$$
C
$$f(x)=x^3 + 7x^2 + 5$$
D
$$f(x)=x^3 - 7x^2 + 15x - 9$$
E
$$f(x)=2x^3 - 3x^2 + 5x + 1$$