Factor Theorem

Front

What is the Factor Theorem?

Back

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0.

Front

How do you determine if (x - 2) is a factor of f(x) = x^3 + 4x^2 + x - 6?

Back

Evaluate f(2). If f(2) = 0, then (x - 2) is a factor. Here, f(2) = 20, so (x - 2) is not a factor.

Front

What does it mean if a polynomial has a root at x = c?

Back

It means that (x - c) is a factor of the polynomial.

Front

What is a sign chart used for?

Back

A sign chart is used to determine the intervals where a polynomial is positive or negative.

Front

If f(x) = (x - 4)(x - 1)^2(x + 2)(x + 4), what are the roots?

Back

The roots are x = 4, x = 1 (with multiplicity 2), x = -2, and x = -4.

Front

What is the end behavior of a polynomial?

Back

The end behavior describes how the values of the polynomial behave as x approaches positive or negative infinity.

Front

True or False: If f(4) ≠ 0, then (x - 4) is not a factor of f(x).

Back

True.

Front

What is the significance of multiplicity in polynomial roots?

Back

Multiplicity indicates how many times a root is repeated. A root with odd multiplicity crosses the x-axis, while even multiplicity touches the x-axis.

Front

How can you find the factored form of a polynomial?

Back

You can find the factored form by identifying its roots and writing it as a product of factors corresponding to those roots.

Front

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

Back

A polynomial of degree n can have at most n roots, counting multiplicities.