Honors Algebra 2 Theorems Flashcards

Questions and Answers

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?

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.

True

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) = ______.

0 Signup and view all the answers

What does the Remainder Theorem explain?

The remainder of f(x) divided by x - k is f(k). Signup and view all the answers

What is the statement of the Factor Theorem?

If f(k) = 0, then x - k is a factor of f(x). Signup and view all the answers

What form do rational solutions take according to the Rational Root Theorem?

They are of the form p/q. Signup and view all the answers

Descarte's Rule of Signs states that only even degree terms change the sign when -x is plugged in.

False Signup and view all the answers

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.

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.