## Table of Contents

- What is the Fundamental Theorem of Algebra
- The Roots of a Polynomial
- Fundamental Theorem of Algebra Proof
- Lesson Summary

- What is the Fundamental Theorem of Algebra
- The Roots of a Polynomial
- Fundamental Theorem of Algebra Proof
- Lesson Summary

The definition of the **fundamental theorem of algebra** is: any polynomial of degree n has n roots. It also states that any polynomial has at least one solution. The theorem does not reveal what the roots of a polynomial are, but it does reveal how many there are. It takes basic mathematical skills to solve what the roots are; the theorem only states how many roots are in a polynomial. It also assists in solving polynomial equations, such as x^2-9=0. In the equation, x^2-9=0, use the definition of the fundamental theorem of algebra to find the number of roots.

x^2-9=0

add 9 to both sides

x^2=9

take the square root of both sides

x=\sqrt{9}

since 9 has a square root the solution is

x= +3,-3

The fundamental theorem of algebra says that any polynomial with n degree has n roots. This polynomial had 2 degrees and it was solved to show it has 2 roots.

Polynomial is derived from the Greek words "poly" meaning many and "nominal" meaning terms. A **polynomial** is a mathematical expression that contains two or more algebraic expressions such as constants, variables, and exponents that are combined by addition, subtraction, multiplication, and division. **Polynomial degrees** are defined as being the highest power of a variable in the equation or expression. For example, if a polynomial expression is 7x^5 + 3x^2 +4, the polynomial degree would be 5, since it is the highest degree throughout the entire polynomial. **Polynomial roots or polynomial zeros** are numbers that result in a polynomial function equaling zero. For example, a polynomial function is P(x)= 3x^2 +4x+1

P(-1/3)= 3(-1/3)^2+4(-1/3)+1

= 3(1/9)-4/3+1

=3/9-4/3+1

=3/9-12/9+9/9

=0

P(-1)=3(-1)^2+4(-1)+1

=3-4+1

=0

Graph the polynomial: P(x)=3x^2+4x+1

The red circles on the graph point out the two roots of the polynomial. By the fundamental theorem of algebra, it is understood that by looking at the function, it will have 2 roots. Using basic math skills and setting the function equal to zero, we were able to find its polynomial roots or polynomial zeros. On the graph, its zeros are where the polynomial intersects with the x axis.

A **real number** is any number that can be found on the number line going towards infinity. Examples of real numbers are whole numbers such as 0, -2, or 4, decimals such as 6.3, fractions such as -2/5, or even square roots such as \sqrt{14}. It should be noted that infinity is not a real number and neither is *i*, which is \sqrt{-1}. In the case of *i*, the reason it is not considered a real number is because it is an imaginary number. The \sqrt{-1} does not exist because there is no number whose square is equal to -1. Because of that, *i* is not regarded as a real number, but as a complex number instead. **Complex numbers** are any numbers that include the imaginary number *i*. Some examples of complex numbers are 5*i*, 2+7*i*, and 9.4*i*. Actually, real numbers can be considered a subset of complex numbers because a real number could be written as r + s*i*, where r represents the real number and s represents the coefficient to the imaginary number. So, the real number 5 could be written as a complex number: 5 + 0*i*.

Because the fundamental theorem of algebra states that any polynomial of a degree n has n number of roots, it also means that a polynomial function has at least one root based on the complex number system. Roots of polynomials can be real or complex. For example, if the polynomial function is something like x^2+9, its 2 roots would be +3 and -3, which are real numbers. However, if the polynomial function is something like x^5 - x^4 + x^3 - x^2 - 12x + 12:

x^5 - x^4 + x^3 - 12x + 12 = 0

(x - 1)(x^4 + x^2 - 12) = 0 Factor out the equation

(x - 1)(x^2 - 3)(x^2 + 4) = 0

(x - 1)(x + \sqrt{3})(x - \sqrt{3})(x^2 + 4) = 0

(x - 1)(x + \sqrt{3})(x - \sqrt{3})(x + 2*i*)(x - 2*i*) = 0

This polynomial has 5 roots and 3 of them are real and 2 are complex.

A complex number comes in pairs due to the **Complex Conjugate Root Theorem**. It states that if one complex root is discovered, then its complex conjugate is also a root. This theorem means that if one complex root of a polynomial is a+bi then another root will be a-bi. If a complex root is given to a 3 root polynomial, then the second root is automatically known due to the complex conjugate root theorem.

Here is a table of degrees of Polynomials and their roots:

Degrees of Polynomials | Number of Roots |

1 | 1 real |

2 | 2 real or 2 complex |

3 | 3 real or 1 real and 2 complex |

4 | 4 real or 2 real and 2 complex or 4 complex |

When a function of a polynomial has the same solution twice, it has a **repeated root**. On a graph, a root is repeated when it touches or is tangent to the x axis; it does not pass through the x axis. For example, take the polynomial function:

P(x) = (x -1)^2 (x + 4)

(x -1)^2 (x + 4) = 0

(x + 4) = 0

X = -4

(x - 1)^2 = 0

Because it is squared, the solution to (x - 1)^2 = 0, which is x = 1, occurs twice and is considered a double root.

On the graph, the one root, x = -4, passes through the x axis because it is a single, real root. However, the other solution of x = 1, which is a repeated root, does not pass through the x axis, it just touches it. When a polynomial is graphed, the roots that pass through the x axis are single, individual roots, the roots that are tangent to the x axis are repeated roots.

Let the function be P(x) = x^3 + 3x^2 - 4x

Using the fundamental theorem of algebra definition, any polynomial of degree n has n roots. x^3 is the variable with the highest power, so based on the fundamental theorem of algebra, this equation has 3 roots.

Simplify the equation by factoring:

x^3 + 3x^2 - 4x = 0

x(x^2 + 3x -4) = 0

x = 0

(x^2 + 3x - 4) = 0

(x + 4)(x - 1) = 0

x + 4 = 0

x = - 4

(x-1) = 0

x = 1

The real roots of this polynomial are x = 0,1,4

Here is a graph of the polynomial.

The points circled on the graph are the 3 real roots of the polynomial because they cross through the x axis.

Let the function be P(x) = x^2 + 5x +7

Using the fundamental theorem of algebra's definition, any polynomial of degree n has n roots. x^2 is the variable with the highest power, so based on the fundamental theorem of algebra, this equation has 2 roots.

Simplify the equation by using the quadratic formula:

Quadratic formula: (-b +/- \sqrt{b^2 - 4ac}) \div (2a)

a=1; b=5; c=7

x = -5 +/-( \sqrt{(5^2) - 4(1)(7)}) \div (2(1))

x = -5 +/- (\sqrt{25 -28}) \div (2)

x = -5 +/- (\sqrt{-3}) \div (2)

x = (-5 + 3i) \div (2); (-5 - 3i) \div (2)

The complex roots of the polynomial are (-5 + 3i) \div (2) and (-5 - 3i) \div (2)

Here is a graph of the polynomial:

The graph shows that the function is a polynomial because of its parabola form. It also displays that the function has zero real roots. It has complex roots because the function does not pass through or touch the x axis.

Let the polynomial equation be: P(x) = (x - 4)^2(x + 4)^2

Using the definition of fundamental theorem of algebra, any polynomial of degree n has n roots. Both parts of the polynomial equation have the power 2, so based on the fundamental theorem of algebra, this equation has 2 roots.

(x - 4)^2(x + 4)^2 = 0

(x - 4)^2= 0

x -4 = 0

x = 4

(x + 4)^2 = 0

x + 4 = 0

x = -4

Because both parts of the polynomial are squared, the solution to (x - 4)^2(x + 4)^2 = 0, which is x = -4,4 occur twice and are considered a double roots.

On the graph, the double roots do not pass through the x-axis, the are tangent to it. Therefore, proving the solutions -4 and 4 to be double roots.

The **fundamental theorem of algebra proof** involves another **algebraic theorem**: the linear factorization theorem. According to the factor theorem, when a polynomial f(x) is divided by (x - g), and (x - g) is a factor of f(x), the remainder is simply 0. As a result, if the residual of a division like the ones given above equals zero, (x - g) must be a factor, according to this theorem.

By using the linear factorization theorem, allow a polynomial function, P, be written as P(x) = ax^n+ax^n-1+...a^0. Because of the definition of the linear factorization theorem, the polynomial can be factored as P(x) = (x - g_1)(x - g_2)....(x - g_n) where g_1, g_2, â€¦ g_n are complex numbers. Thus, any polynomial has exactly n solutions among the complex numbers. Therefore, proving the fundamental theorem of algebra.

The **fundamental theorem of algebra** is defined as any polynomial with degree n has n roots. **Polynomials** are exponents that are combined by addition, subtraction, multiplication, and division. The roots of polynomials can be complex or real solutions. **Real numbers **are positive and negative whole numbers, like 4, decimals, fractions, and roots. **Complex solutions** involve imaginary numbers. Real solutions are subsets of complex solutions because real solutions can be written r + s"i". Real solutions pass through the x axis on a graph. Complex solutions do not pass through the x axis and are difficult to be traced on a graph. **Repeated roots** occur when the solution is tangent to the x-axis and occurs twice to solve the polynomial. An algebraic theorem, the **complex conjugate theorem**, says that if one complex root is discovered, then its complex conjugate is also a root. Use basic algebraic skills, other theorems, and algebraic equations, such as the quadratic equation, to solve for roots by hand. The **fundamental theorem of algebra proof** can be found by using the linear factorization theorem. This theorem will assist in demonstrating that the fundamental theorem of algebra works and can be trusted.

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Frequently Asked Questions

You prove the fundamental theorem of algebra by using the linear factorization theorem. By using that theorem, then allow a polynomial function, P, be written as P(x) = ax^n+ax^n-1+...a^0. Because of the definition of the linear factorization theorem, the polynomial can be factored as P(x) = (x - g_1)(x - g_2)....(x - g_n) where g_1, g_2, … g_n are complex numbers. Thus, any polynomial has exactly n solutions among the complex numbers.

The fundamental theorem of algebra is used in real life as a structure for other algebraic and trigonometric areas of study. It is used also in linear algebra and algebraic geometry.

The fundamental theorem of algebra states that any polynomial of degree n has n roots. It also states that any polynomial has at least one solution. The theorem does not reveal what the roots of a polynomial are, but it does reveal how many there are.

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