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 is f(c).

Front

How do you find the remainder of a polynomial when divided by a linear factor?

Back

To find the remainder of a polynomial f(x) when divided by (x - c), evaluate f(c).

Front

If f(x) = 2x^3 - 3x + 5, what is the remainder when f(x) is divided by (x - 1)?

Back

The remainder is f(1) = 2(1)^3 - 3(1) + 5 = 4.

Front

Is (x + 1) a factor of f(x) = x^2 - 3x + 2?

Back

Yes, because f(-1) = 0.

Front

What does it mean if the remainder is 0 when dividing a polynomial by a linear factor?

Back

It means that the linear factor is a factor of the polynomial.

Front

If f(x) = x^3 - 4x + 1, what is the remainder when divided by (x - 2)?

Back

The remainder is f(2) = 2^3 - 4(2) + 1 = -1.

Front

How can you determine if (x - c) is a factor of f(x)?

Back

Evaluate f(c). If f(c) = 0, then (x - c) is a factor.

Front

What is the significance of the coefficients in a polynomial?

Back

The coefficients determine the shape and position of the polynomial's graph.

Front

If f(x) = 3x^4 - 5x^3 + 2, what is the remainder when divided by (x + 2)?

Back

The remainder is f(-2) = 3(-2)^4 - 5(-2)^3 + 2 = 50.