Remainder theorem/Factor theorem

Front

What is the Remainder Theorem?

Back

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

Front

What is the Factor Theorem?

Back

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

Front

If f(x) = x^2 + 3x + 7, what is f(-2)?

Back

f(-2) = (-2)^2 + 3(-2) + 7 = 4 - 6 + 7 = 5.

Front

How do you find the remainder when dividing polynomials?

Back

To find the remainder, evaluate the polynomial at the value that makes the divisor zero.

Front

Is (x - 7) a factor of (x^2 - 5x + 14)?

Back

No, because f(7) = 7^2 - 5(7) + 14 = 49 - 35 + 14 = 28, which is not zero.

Front

What is the remainder when (3x^3 - 5x^2 - 4x + 1) is divided by (x + 3)?

Back

The remainder is -113.

Front

What does it mean if a polynomial has a remainder of 0 when divided by (x - c)?

Back

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

Front

If f(x) = x^3 - 3x^2 + 2, what is f(1)?

Back

f(1) = 1^3 - 3(1)^2 + 2 = 1 - 3 + 2 = 0.

Front

What is synthetic division?

Back

Synthetic division is a simplified form of polynomial long division that is used when dividing by a linear factor.

Front

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

Back

The leading coefficient affects the end behavior of the polynomial and can influence the degree of the quotient.