PreCalculus Flashcard 2.4

Front

What are the zeros of a polynomial function?

Back

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

Front

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

Back

To find the zeros of a polynomial given one factor, set the factor equal to zero and solve for x. For example, if (x - 3) is a factor, set x - 3 = 0, which gives x = 3 as a zero.

Front

What does it mean if P(c) = 0 for a polynomial P(x)?

Back

If P(c) = 0, then x - c is a factor of the polynomial P(x). This means that c is a root or zero of the polynomial.

Front

How many total roots can a polynomial of degree n have?

Back

A polynomial of degree n can have up to n roots, counting multiplicities.

Front

What is the relationship between the factors and zeros of a polynomial?

Back

The factors of a polynomial correspond to its zeros. If x - r is a factor, then r is a zero of the polynomial.

Front

What is synthetic division and when is it used?

Back

Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor of the form (x - c). It is used to find zeros and simplify polynomials.

Front

What is the Factor Theorem?

Back

The Factor Theorem states that if P(c) = 0 for a polynomial P(x), then (x - c) is a factor of P(x).

Front

What is the Remainder Theorem?

Back

The Remainder Theorem states that when a polynomial P(x) is divided by (x - c), the remainder is P(c).

Front

How can you determine if (x - r) is a factor of P(x)?

Back

To determine if (x - r) is a factor of P(x), evaluate P(r). If P(r) = 0, then (x - r) is a factor.

Front

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

Back

The leading coefficient of a polynomial determines the end behavior of the graph. It influences whether the graph rises or falls as x approaches positive or negative infinity.