Factor and Remainder 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

What is the Remainder Theorem?

Back

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

Front

How do you perform synthetic division?

Back

Synthetic division is a simplified form of polynomial long division. It involves using the coefficients of the polynomial and the value of c to find the quotient and remainder.

Front

What is the result of dividing (2x^3 + 5x^2 + 9) by (x + 3)?

Back

The result is 2x^2 - x + 3.

Front

Is (x - 3) a factor of (x^3 - 3x^2 + 2x + 2)?

Back

No, (x - 3) is not a factor.

Front

What is the value of f(-3) for f(x) = (3x^3 - 5x^2 - 4x + 1) using synthetic division?

Back

The value is -113.

Front

How do you find f(5) for f(x) = 3x^2 - 9x - 20 using synthetic division?

Back

By substituting x = 5, the value is 10.

Front

What does it mean if a polynomial has a factor?

Back

It means that the polynomial can be divided by that factor without leaving a remainder.

Front

What is the significance of the roots of a polynomial?

Back

The roots of a polynomial are the values of x for which the polynomial equals zero, and they correspond to the factors of the polynomial.

Front

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

Back

If (x - c) is a factor of a polynomial, then c is a zero (or root) of that polynomial.