PC Ch 2 - Zeros of Polynomial Functions

Front

What is a zero of a polynomial function?

Back

A zero of a polynomial function is a value of x for which the polynomial evaluates to zero, meaning P(x) = 0.

Front

How do you find the zeros of a polynomial function?

Back

To find the zeros of a polynomial function, you can use methods such as factoring, synthetic division, or applying the Rational Root Theorem.

Front

What is synthetic division?

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).

Front

What is the Fundamental Theorem of Algebra?

Back

The Fundamental Theorem of Algebra states that every non-constant polynomial function has at least one complex root, and the total number of roots (including multiplicities) is equal to the degree of the polynomial.

Front

What are complex roots?

Back

Complex roots are solutions to polynomial equations that include imaginary numbers, typically in the form a + bi, where a and b are real numbers.

Front

If a polynomial has real coefficients, what can be said about its complex roots?

Back

If a polynomial has real coefficients, then any complex roots must occur in conjugate pairs, meaning if a + bi is a root, then a - bi is also a root.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial expression.

Front

How do you determine the number of possible real zeros of a polynomial?

Back

The number of possible real zeros of a polynomial can be determined by its degree; a polynomial of degree n can have up to n real zeros.

Front

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

Back

The zeros of a polynomial correspond to the values of x that make each factor equal to zero, meaning if (x - r) is a factor, then r is a zero.

Front

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

Back

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