Podcast
Questions and Answers
What does the Complex Conjugate Theorem state?
What does the Complex Conjugate Theorem state?
- Every polynomial has roots.
- If a + bi is a root, then a - bi is also a root. (correct)
- All roots must be real numbers.
- Imaginary roots don't exist.
What does the Irrational Root Theorem state?
What does the Irrational Root Theorem state?
If a + √b is a root, then a - √b is also a root.
The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots.
The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots.
True (A)
According to the Intermediate Value Theorem, if f(a) and f(b) have opposite signs, then there is at least one value of c between a and b for which f(c) = ______.
According to the Intermediate Value Theorem, if f(a) and f(b) have opposite signs, then there is at least one value of c between a and b for which f(c) = ______.
What does the Remainder Theorem explain?
What does the Remainder Theorem explain?
What is the statement of the Factor Theorem?
What is the statement of the Factor Theorem?
What form do rational solutions take according to the Rational Root Theorem?
What form do rational solutions take according to the Rational Root Theorem?
Descarte's Rule of Signs states that only even degree terms change the sign when -x is plugged in.
Descarte's Rule of Signs states that only even degree terms change the sign when -x is plugged in.
Study Notes
Complex Conjugate Theorem
- If a polynomial equation has real coefficients and includes an imaginary root a + bi, the conjugate a - bi must also be a root.
Irrational Root Theorem
- For polynomial equations with rational coefficients, if a root is of the form a + √b (where √b is irrational), then its conjugate a - √b is also a root.
Fundamental Theorem of Algebra
- Every polynomial of degree n (n ≥ 1) with complex coefficients has at least one root in the complex number system.
- An nth degree polynomial has exactly n roots, accounting for both real and complex (including multiple) roots.
Intermediate Value Theorem
- In a polynomial function with real coefficients, if f(a) and f(b) have opposite signs, there exists at least one value c in the interval (a, b) such that f(c) = 0, indicating a root lies between a and b.
Remainder Theorem
- When dividing a polynomial f(x) by x - k, the remainder of this division is equal to f(k).
Factor Theorem
- A polynomial f(x) has a factor x - k if and only if f(k) = 0; conversely, if x - k is a factor, it confirms that f(k) = 0.
Rational Root Theorem
- Any rational solution of a polynomial f(x) can be expressed in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
Descartes' Rule of Signs
- In a polynomial, the number of sign changes when substituting -x indicates the possible number of positive real roots; only terms of odd degree contribute to sign changes.
Studying That Suits You
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Description
Test your understanding of essential theorems in Honors Algebra 2 with these flashcards. Each card presents a theorem along with its definition, helping you to memorize the relationships and properties of polynomial roots. Ideal for students preparing for exams or wishing to reinforce their algebra knowledge.
