PreCalculus Flashcard 2.4

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

How do you find the zeros of a polynomial function?

Back

To find the zeros of a polynomial function, set the function equal to zero and solve for x. This can involve factoring, using the quadratic formula, or synthetic division.

Front

What is the Rational Root Theorem?

Back

The Rational Root Theorem states that any rational solution (or zero) 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

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 of the form (x - c). It is quicker and requires less writing.

Front

What is a polynomial?

Back

A polynomial is a mathematical expression consisting of variables (x) raised to non-negative integer powers and coefficients, combined using addition, subtraction, and multiplication.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial. It determines the polynomial's behavior and the maximum number of zeros it can have.

Front

What does it mean for a polynomial to have multiple roots?

Back

A polynomial has multiple roots when a particular root occurs more than once. For example, in (x - 2)², the root x = 2 is a double root.

Front

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

Back

The leading coefficient is the coefficient of the term with the highest degree. It affects the end behavior of the polynomial graph.

Front

How can you determine the number of real roots of a polynomial?

Back

The number of real roots can be determined using the Fundamental Theorem of Algebra, which states that a polynomial of degree n has exactly n roots, counting multiplicities.

Front

What is the difference between rational and irrational roots?

Back

Rational roots can be expressed as a fraction of two integers, while irrational roots cannot be expressed as a simple fraction and often involve square roots or other non-repeating decimals.