Algebra II 4.09: Fundamental Theorem of Algebra

Questions and Answers

How many complex roots does the polynomial equation $3x^5−2^x+1=0$ have?

Which answer best describes the complex zeros of the polynomial function $f(x)=x^3+x^2+10x+10$?

The function has one real zero and two nonreal zeros. The graph of the function intersects the x-axis at exactly one location.

How many complex roots does the equation $0=3x^5−x^4−5x^2+1$ have?

What is the completely factored form of $f(x)=x^3+5x^2+4x−6$?

f(x)=(x+3)(x−(−1+3‾√))(x−(−1−3‾√3)) Signup and view all the answers

What is the complete factorization of the polynomial function over the set of complex numbers for $f(x)=x^3−4x^2+4x−16$?

(x-4)(x+2i)(x-2i) Signup and view all the answers

How many complex roots does the polynomial equation $5x^3−4x+1=0$ have?

3 Signup and view all the answers

Which answer best describes the complex zeros of the polynomial function $f(x)=x^3−x^2+6x−6$?

The function has one real zero and two nonreal zeros. The graph of the function intersects the x-axis at exactly one location. Signup and view all the answers

How many complex roots does the equation $0=4x^4−x^3−5x+3$ have?

4 Signup and view all the answers

What is the completely factored form of $f(x)=x^3+4x^2+7x+6$?

f(x)=(x+2)(x−(−1+i√2))(x−(−1−i√2)) Signup and view all the answers

What is the complete factorization of the polynomial function over the set of complex numbers for $f(x)=x^3+3x^2+16x+48$?

(x+3)(x-4i)(x+4i) Signup and view all the answers

Study Notes

Complex Roots in Polynomial Equations

A polynomial of degree five, such as (3x^5−2^x+1=0), will have exactly five complex roots including real numbers, if any.
The polynomial (3x^5−x^4−5x^2+1=0) also yields five complex roots as it is a degree five polynomial.

Complex Zeros of Specific Functions

The function (f(x)=x^3+x^2+10x+10) has one real zero and two nonreal zeros, indicated by its intersection at a single point on the x-axis.
Similarly, (f(x)=x^3−x^2+6x−6) displays the same characteristics with one real root and two nonreal zeros.

Factorization of Polynomial Functions

The completely factored form of (f(x)=x^3+5x^2+4x−6) is given as (f(x)=(x+3)(x−(−1+\sqrt{3}))(x−(−1−\sqrt{3}))).
For (f(x)=x^3−4x^2+4x−16), the complete factorization yields ( (x-4)(x+2i)(x-2i) ), indicating the presence of complex conjugates (2i) and (-2i).

Degree and Roots Relationship

The polynomial (5x^3−4x+1=0) is cubic and thus has three complex roots.
(4x^4−x^3−5x+3=0), being a quartic polynomial, will contain four complex roots total.

Complex Conjugates in Polynomials

For (f(x)=x^3+4x^2+7x+6), the factored form includes complex conjugates (-1+i\sqrt{2}) and (-1-i\sqrt{2}).
In the case of (f(x)=x^3+3x^2+16x+48), the factors include (x-4i) and (x+4i), demonstrating the relationship of complex roots appearing in conjugate pairs in polynomials with real coefficients.

Description

Explore the Fundamental Theorem of Algebra through flashcards that cover complex roots and zeros of polynomial equations. This quiz provides essential insights into calculating and understanding complex numbers in algebraic functions.