Zeros of Polynomials

Front

What does the Fundamental Theorem of Algebra state?

Back

Every non-constant polynomial equation of degree n has exactly n complex roots, counting multiplicities.

Front

Define a zero of a polynomial.

Back

A zero of a polynomial is a value of x that makes the polynomial equal to zero.

Front

What is the Rational Root Theorem?

Back

The Rational Root Theorem states that any rational solution (or root) of a polynomial equation can be expressed as a fraction p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

Front

How do you find the zeros of a polynomial given one zero?

Back

Use synthetic division or polynomial long division to divide the polynomial by (x - zero), then find the zeros of the resulting polynomial.

Front

What is the difference between real and complex zeros?

Back

Real zeros are the x-intercepts of the polynomial on the graph, while complex zeros are solutions that involve imaginary numbers and do not correspond to x-intercepts.

Front

What is a multiplicity of a zero?

Back

The multiplicity of a zero refers to the number of times a particular zero is repeated as a root of the polynomial.

Front

How can you verify if a number is a zero of a polynomial?

Back

Substitute the number into the polynomial; if the result is zero, then it is a zero of the polynomial.

Front

What is synthetic division?

Back

Synthetic division is a simplified form of polynomial long division that is used to divide a polynomial by a linear factor.

Front

What are the possible rational zeros of the polynomial x^3 - 6x^2 + 11x - 6?

Back

The possible rational zeros are ±1, ±2, ±3, ±6.

Front

What is the significance of the leading coefficient in a polynomial?

Back

The leading coefficient determines the end behavior of the polynomial's graph.