Understand what the definition of fundamental theorem of algebra is. See the proof for the fundamental theorem of algebra with examples`of algebra theorems. Updated: 01/11/2022 What is the Fundamental Theorem of AlgebraThe 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. Bank Fee AnalogyThis lesson will show you how to interpret the fundamental theorem of algebra. After completing this lesson, you will be able to state the theorem and explain what it means. Before we state the theorem, we will consider the following analogy. Let's say your bank charges a fee every time you withdraw money from an automatic teller machine. If you withdraw money five times in a particular month, then you will expect five respective bank fees on that month's statement. Let's change this statement by using some mathematical lingo: If you withdraw money n times in a particular month, then you will expect n respective bank fees on that month's statement. 5 withdrawals = 5 bank fees The fundamental theorem of algebra is just as straightforward as this banking analogy.
The Roots of a PolynomialPolynomial 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. Real and Complex RootsA 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 5i, 2+7i, and 9.4i. Actually, real numbers can be considered a subset of complex numbers because a real number could be written as r + si, 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 + 0i. 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 + 2i)(x - 2i) = 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:
Repeated RootsWhen 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. Fundamental Theorem of Algebra Example 1Let 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 Fundamental Theorem of AlgebraThe fundamental theorem of algebra states the following: A polynomial function f(x) of degree n (where n > 0) has n complex solutions for the equation f(x) = 0. Please note that the terms 'zeros' and 'roots' are synonymous with solutions as used in the context of this lesson. That is pretty much it. Now, we should already know that polynomials can be described by their degree. For example, the polynomial x^3 + 3x^2 - 6x - 8 has a degree of 3 because its largest exponent is 3. The degree of a polynomial is important because it tells us the number of solutions of a polynomial. The theorem does not tell us what the solutions are. It only tells us how many solutions exist for a given polynomial function. So what good is that? First of all, it is important to understand underlying concepts of any math topics you are learning. In addition, the fundamental theorem of algebra has practical applications. For instance, if you need to find the solutions of a polynomial function, say, of degree 4, you know that you need to keep working until you find 4 solutions. Imaginary SolutionsIt is important to note that the theorem says complex solutions, so some solutions might be imaginary or have an imaginary part. Maybe we should do a quick review of complex numbers. Complex numbers are in the form of a + bi (a and b are real numbers). The term a is the real part, and the term bi is the imaginary part. If b = 0, then the number is a real number. Therefore, all real numbers are complex numbers. Let's look at a couple of examples: In the complex number 2 + 3i, 2 is the real part and 3i is the imaginary part. In the complex number 25 + 0i, 25 is the real part and 0i is the imaginary part. Because b = 0, the number simplifies to 25. Repeated SolutionsBefore we look at some examples of polynomial functions, let's clarify the concept of repeated solutions. A polynomial function has repeated solutions if it has repeated factors. A good way to show this is with the function f(x) = x^3. This function has a degree of 3, so based on our theorem, it has 3 solutions. We might see the three solutions better if we show the function in factored form: f(x) = (x)(x)(x). Let's now make the function equal to zero: 0 = (x)(x)(x). If any of the three factors equal zero, then the function equals zero. Therefore, the solutions are x = 0, x = 0, and x = 0. The solution of zero occurs 3 times. Example #1Let's start with the polynomial function f(x) = x^2 + 9. In factored form, this function equals (x - 3i)(x + 3i). Bank Fee AnalogyThis lesson will show you how to interpret the fundamental theorem of algebra. After completing this lesson, you will be able to state the theorem and explain what it means. Before we state the theorem, we will consider the following analogy. Let's say your bank charges a fee every time you withdraw money from an automatic teller machine. If you withdraw money five times in a particular month, then you will expect five respective bank fees on that month's statement. Let's change this statement by using some mathematical lingo: If you withdraw money n times in a particular month, then you will expect n respective bank fees on that month's statement. 5 withdrawals = 5 bank fees The fundamental theorem of algebra is just as straightforward as this banking analogy. Fundamental Theorem of AlgebraThe fundamental theorem of algebra states the following: A polynomial function f(x) of degree n (where n > 0) has n complex solutions for the equation f(x) = 0. Please note that the terms 'zeros' and 'roots' are synonymous with solutions as used in the context of this lesson. That is pretty much it. Now, we should already know that polynomials can be described by their degree. For example, the polynomial x^3 + 3x^2 - 6x - 8 has a degree of 3 because its largest exponent is 3. The degree of a polynomial is important because it tells us the number of solutions of a polynomial. The theorem does not tell us what the solutions are. It only tells us how many solutions exist for a given polynomial function. So what good is that? First of all, it is important to understand underlying concepts of any math topics you are learning. In addition, the fundamental theorem of algebra has practical applications. For instance, if you need to find the solutions of a polynomial function, say, of degree 4, you know that you need to keep working until you find 4 solutions. Imaginary SolutionsIt is important to note that the theorem says complex solutions, so some solutions might be imaginary or have an imaginary part. Maybe we should do a quick review of complex numbers. Complex numbers are in the form of a + bi (a and b are real numbers). The term a is the real part, and the term bi is the imaginary part. If b = 0, then the number is a real number. Therefore, all real numbers are complex numbers. Let's look at a couple of examples: In the complex number 2 + 3i, 2 is the real part and 3i is the imaginary part. In the complex number 25 + 0i, 25 is the real part and 0i is the imaginary part. Because b = 0, the number simplifies to 25. Repeated SolutionsBefore we look at some examples of polynomial functions, let's clarify the concept of repeated solutions. A polynomial function has repeated solutions if it has repeated factors. A good way to show this is with the function f(x) = x^3. This function has a degree of 3, so based on our theorem, it has 3 solutions. We might see the three solutions better if we show the function in factored form: f(x) = (x)(x)(x). Let's now make the function equal to zero: 0 = (x)(x)(x). If any of the three factors equal zero, then the function equals zero. Therefore, the solutions are x = 0, x = 0, and x = 0. The solution of zero occurs 3 times. Example #1Let's start with the polynomial function f(x) = x^2 + 9. In factored form, this function equals (x - 3i)(x + 3i). How do you prove the fundamental theorem of algebra?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. What is the fundamental theorem of algebra used for in real life?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. What is the fundamental theorem of algebra?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. Register to view this lessonAre you a student or a teacher? Unlock Your EducationSee for yourself why 30 million people use Study.comBecome a Study.com member and start learning now.Become a Member Already a member? Log In Back Resources created by teachers for teachersOver 30,000 video lessons & teaching resources‐all in one place. Video lessons Quizzes & Worksheets Classroom Integration Lesson Plans I would definitely recommend Study.com to my colleagues. It’s like a teacher waved a magic wand and did the work for me. I feel like it’s a lifeline. Back Create an account to start this course today Used by over 30 million students worldwide Create an account How many roots does fundamental theorem of algebra?The Fundamental Theorem of Algebra states that this 2 n d degree polynomial must have exactly 2 roots with multiplicity.
What is the fundamental theorem of algebra for polynomials?fundamental theorem of algebra, theorem of equations proved by Carl Friedrich Gauss in 1799. It states that every polynomial equation of degree n with complex number coefficients has n roots, or solutions, in the complex numbers. The roots can have a multiplicity greater than zero.
What formula is the fundamental theorem of algebra?The fundamental theorem of algebra states the following: A polynomial function f(x) of degree n (where n > 0) has n complex solutions for the equation f(x) = 0. Please note that the terms 'zeros' and 'roots' are synonymous with solutions as used in the context of this lesson. That is pretty much it.
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According to the fundamental theorem of algebra which polynomial function has exactly 6 roots
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