Finding Zeros of Polynomials Flashcard

Front

What are the zeros of a polynomial?

Back

The zeros of a polynomial are the values of x for which the polynomial evaluates to zero. They are also known as the roots of the polynomial.

Front

State the Fundamental Theorem of Algebra.

Back

The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n complex roots, counting multiplicities.

Front

How do you find the zeros of a polynomial function?

Back

To find the zeros of a polynomial function, set the polynomial equal to zero and solve for x using factoring, the quadratic formula, or numerical methods.

Front

What is a rational zero?

Back

A rational zero is a solution to a polynomial equation that 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

What is the relationship between the degree of a polynomial and the number of zeros?

Back

The degree of a polynomial indicates the maximum number of zeros it can have. A polynomial of degree n can have up to n zeros.

Front

What is synthetic division and how is it used in finding zeros?

Back

Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor (x - c). It helps in finding zeros by testing possible rational roots.

Front

What is the significance of the discriminant in a quadratic equation?

Back

The discriminant (b² - 4ac) determines the nature of the roots of a quadratic equation: if positive, there are two distinct real roots; if zero, one real root; if negative, two complex roots.

Front

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

Back

To verify if a number c is a zero of a polynomial p(x), substitute c into the polynomial. If p(c) = 0, then c is a zero.

Front

What is the Rational Root Theorem?

Back

The Rational Root Theorem provides a way to find all possible rational roots of a polynomial equation by listing the factors of the constant term and the leading coefficient.

Front

What does it mean for a zero to have multiplicity?

Back

A zero has multiplicity if it is a root of the polynomial more than once. For example, if (x - 2)² is a factor, then x = 2 is a zero with multiplicity 2.