### Learning Outcomes

- Use synthetic division to find the zeros of a polynomial function.
- Use the Fundamental Theorem of Algebra to find complex zeros of a polynomial function.

The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use **synthetic division** repeatedly to determine all of the **zeros** of a polynomial function.

### How To: Given a polynomial function [latex]f[/latex], use synthetic division to find its zeros

- Use the Rational Zero Theorem to list all possible rational zeros of the function.
- Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate.
- Repeat step two using the quotient found from synthetic division. If possible, continue until the quotient is a quadratic.
- Find the zeros of the quadratic function. Two possible methods for solving quadratics are factoring and using the quadratic formula.

### Example: Finding the Zeros of a Polynomial Function with Repeated Real Zeros

Find the zeros of [latex]f\left(x\right)=4{x}^{3}-3x - 1[/latex].

## The Fundamental Theorem of Algebra

Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function. The **Fundamental Theorem of Algebra **tells us that every polynomial function has at least one complex zero. This theorem forms the foundation for solving polynomial equations.

Suppose *f* is a polynomial function of degree four and [latex]f\left(x\right)=0[/latex]. The Fundamental Theorem of Algebra states that there is at least one complex solution, call it [latex]{c}_{1}[/latex]. By the Factor Theorem, we can write [latex]f\left(x\right)[/latex] as a product of [latex]x-{c}_{\text{1}}[/latex] and a polynomial quotient. Since [latex]x-{c}_{\text{1}}[/latex] is linear, the polynomial quotient will be of degree three. Now we apply the Fundamental Theorem of Algebra to the third-degree polynomial quotient. It will have at least one complex zero, call it [latex]{c}_{\text{2}}[/latex]. We can write the polynomial quotient as a product of [latex]x-{c}_{\text{2}}[/latex] and a new polynomial quotient of degree two. Continue to apply the Fundamental Theorem of Algebra until all of the zeros are found. There will be four of them and each one will yield a factor of [latex]f\left(x\right)[/latex].

### A General Note: The **Fundamental Theorem of Algebra**

The Fundamental Theorem of Algebra states that, if [latex]f(x)[/latex] is a polynomial of degree [latex]n>0[/latex], then [latex]f(x)[/latex] has at least one complex zero.

We can use this theorem to argue that, if [latex]f\left(x\right)[/latex] is a polynomial of degree [latex]n>0[/latex], and *a* is a non-zero real number, then [latex]f\left(x\right)[/latex] has exactly *n* linear factors.

The polynomial can be written as

[latex]f\left(x\right)=a\left(x-{c}_{1}\right)\left(x-{c}_{2}\right)…\left(x-{c}_{n}\right)[/latex]

where [latex]{c}_{1},{c}_{2},…,{c}_{n}[/latex] are complex numbers. Therefore, [latex]f\left(x\right)[/latex] has *n* roots if we allow for multiplicities.

### recall complex numbers

Recall that we defined complex numbers as numbers of the form [latex]a + bi[/latex]. To graph such numbers required the complex plane, made up of a real axis and an *imaginary* axis. This plane defined the real numbers as a subset of the complex numbers, just as the rational and irrational numbers are subsets of the real numbers.

In other words, each real number is also a complex number of the form [latex]a+bi[/latex], where [latex]b=0[/latex].

### Q & A

**Does every polynomial have at least one imaginary zero?**

*No. A complex number is not necessarily imaginary. Real numbers are also complex numbers.*

### Example: Finding the Zeros of a Polynomial Function with Complex Zeros

Find the zeros of [latex]f\left(x\right)=3{x}^{3}+9{x}^{2}+x+3[/latex].

### Try It

Find the zeros of [latex]f\left(x\right)=2{x}^{3}+5{x}^{2}-11x+4[/latex].