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## The Fundamental Theorem of Algebra

#### High School - HSN-CN.C.9

What is the Fundamental Theorem of Algebra? It is always difficult to understand complex terms and terminologies, particularly those that entail complex zeros and complex roots. It is a theorem that discusses the number of complex zeros of a polynomial function. The Fundamental Theorem of Algebra is a theorem that was first proved by Gauss. Every polynomial has one complex root. It states that every non-constant single-variable polynomial having complex coefficients at least consists of one complex root. The polynomials that it includes may be ones with real coefficients having real and imaginary parts equal to zero. Looking at the theorem, according to the equivalent definition, the theorem represents that it is algebraically close. Consider f is a polynomial function of degree four and f(x) = 0. By the Factor Theorem, we can consider f(x) the product of x – c1. This polynomial is of degree 3. Applying the Fundamental Theorem of Algebra to it, it will have one complex zero, call it. The key is to apply the Fundamental Theorem of Algebra until you find all the zeros in the expression.

### Printable Worksheets And Lessons

• Binomial Roots Step-by-step Lesson- Find all the possible roots of a fixed polynomial.

• Guided Lesson - Three pretty drastically different problems. I had to do a double take to make sure they all fell under this skill.

• Guided Lesson Explanation - Once you see the approaches on this one, it is very understandable.

• Practice Worksheet - One root after another, followed by solving equations, and finding polynomials with given conditions.

• Matching Worksheet - Match the problems to their outcome. This one is funny kids with either get them all right or wrong there is very little in between.  #### Homework Sheets

This work is as I like to call it, "Deep Algebra".

• Homework 1 - We can see that there will be 2 roots because it has a degree of 2.
• Homework 2 - Move the variables to the other side usually with basic operations.
• Homework 3 - We should know that if roots are formed with given conditions, flipping the sign of the roots in relation to x will give a polynomial expressed as a product of three linear factors.

#### Practice Worksheets

Start by determining the roots of an expression. Then we solve and break apart polynomials.

• Practice 1 - What are the roots of x2 - 169?
• Practice 2 - Solve the equation and write any complex solutions in the form a + bi, where a and b are real numbers.
• Practice 3 - Find a polynomial with integer satisfying the following conditions: Degree 3 with roots at 11, 2, and -3.

#### Math Skill Quizzes

Based on teacher request we had a single skill per quiz for this section.

• Quiz 1 - We want it to be equal to zero, this will allow us to find the value.
• Quiz 2 - We will create polynomial, P (x), from this by multiplying the factors.
• Quiz 3 - Find the right conditions for this to fit in there.