Mathematics
Grade 11
15 min
Fundamental Theorem of Algebra
Fundamental Theorem of Algebra
What you'll learn
- Identify whether a multiplication sentence (up to 10 x 10) is true or false by calculating the product.
- Solve multiplication problems (up to 10 x 10) to determine if a given multiplication sentence is correct.
- Explain why a multiplication sentence (up to 10 x 10) is true or false using mathematical reasoning and the concept of equal groups.
- Evaluate different multiplication sentences (up to 10 x 10) and justify your answer of true or false with evidence of your calculations.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
State the Fundamental Theorem of Algebra and its corollary.
Determine the total number of roots for a polynomial by identifying its degree.
Apply the Complex Conjugate Root Theorem to identify pairs of complex roots.
Find all real and complex roots of a polynomial function.
Construct a polynomial function in standard form given its roots.
Understand the concept of multiplicity of roots.
Ever wondered if every polynomial equation has a solution, even if you can't see it on a graph? 🤔 This theorem guarantees it!
The Fundamental Theorem of Algebra is a cornerstone concept that guarantees every non-constant polynomial has at least one root in the complex number system. This lesson will show you how to find every single root, both real and complex, for...
2
Key Concepts & Vocabulary
TermDefinitionExample
Polynomial FunctionA function of the form P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where 'n' is a non-negative integer and the coefficients 'a' are real numbers.P(x) = 3x^4 - 5x^2 + 2x - 7
Degree of a PolynomialThe highest exponent of the variable in a polynomial.The polynomial P(x) = 3x^4 - 5x^2 + 2x - 7 has a degree of 4.
Root (or Zero)A value 'c' for the variable 'x' that makes the polynomial equal to zero, i.e., P(c) = 0. On a graph, real roots are the x-intercepts.For P(x) = x^2 - 4, the roots are x = 2 and x = -2 because P(2) = 0 and P(-2) = 0.
Complex NumberA number written in the form a + bi, where 'a' is the real part, 'b' is the imaginary part, and i is the imaginary unit (i^2 = -1).3...
3
Core Formulas
The Fundamental Theorem of Algebra
Every non-constant single-variable polynomial with complex coefficients has at least one complex root.
This is the core guarantee. It tells us that a solution always exists, as long as we are working within the complex number system. You don't solve with this theorem, but it's the foundation for why we can always find roots.
The n-Roots Corollary
Every polynomial of degree n > 0 has exactly n roots, counting multiplicity, in the set of complex numbers.
This is the practical application of the theorem. It tells you exactly how many roots to look for. A polynomial of degree 3 has 3 roots, a polynomial of degree 5 has 5 roots, and so on.
The Complex Conjugate Root Theorem
If a polynomial P(x) has real coefficients, and if a...
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Challenging
A polynomial P(x) with real coefficients has degree 6. It is known that 2i, 1-i, and -3 are roots. If the root -3 has a multiplicity of 2, what is the complete list of roots?
A.{2i, -2i, 1-i, 1+i, -3, -3}
B.{2i, 1-i, -3, -3, 3, i-1}
C.{2i, -2i, 1-i, -1+i, -3, 3}
D.{2i, -2i, 1-i, 1+i, -3, 3}
Challenging
Let P(x) be a polynomial with real coefficients. If P(a + bi) = 0, where b is a non-zero real number, what is the value of P(a - bi)?
A.1
B.0
C.P(a) + P(bi)
D.It cannot be determined from the information given.
Challenging
A polynomial P(x) = x^4 + ax^2 + b has real coefficients 'a' and 'b'. If one root of the polynomial is 2 - i, what is the value of 'b'?
A.5
B.10
C.20
D.25
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Frequently asked questions
What grade level is "Fundamental Theorem of Algebra"?
Fundamental Theorem of Algebra is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Fundamental Theorem of Algebra?
You'll be able to: Identify whether a multiplication sentence (up to 10 x 10) is true or false by calculating the product; Solve multiplication problems (up to 10 x 10) to determine if a given multiplication sentence is correct; Explain why a….
Is "Fundamental Theorem of Algebra" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Fundamental Theorem of Algebra?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.