MPF - Zeros of Polynomial Function Lesson

Zeros of Polynomial Function

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant single variable polynomial with complex coefficients has at least one complex root. In other words, a polynomial function of degree n has n roots (or solutions), including real and complex roots.

Watch the following videos on Solving by Quadratic Formula with Complex Solutions and the Fundamental Theorem of Algebra.

For Polynomial Functions, the  zeros  are the roots or solutions or x-intercepts (each of these 4 terms refers to the same numbers for a function.) We will learn several ways to find these zeros.

Let's review solving polynomial equations by factoring. There is a very important property we will use called the Zero Product Property which states that: If $ab=0$, then $a=0\:or\:b=0$. This property works only for zero! You can't have two numbers whose product is 5 and assume that one of the numbers is 5 (could be 2.5 times 2)! Again, it's the ZERO product property. To solve these types of problems, you will first get the equation equal to 0, then factor the equation. They will be in the general form $\left(x-a\right)\left(x+b\right)=0$. When using the Zero Product Property, we will take each factor and set it equal to zero to get the answers.

Let's look at some higher order equations (cubic and quartic) that can also be solved by factoring. Remember that we have already looked at solving quadratic polynomials by the factoring method. We have also touched on factoring higher order polynomials, but we will go into more depth now in solving higher order polynomials.

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