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

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

Back

To determine if (x - c) is a factor of f(x), evaluate f(c). If f(c) = 0, then (x - c) is a factor.

Front

Evaluate f(3) for f(x) = 3x^3 + 10x^2 - x - 12.

Back

f(3) = 3(3)^3 + 10(3)^2 - (3) - 12 = 81 + 90 - 3 - 12 = 156.

Front

Is (x - 3) a factor of f(x) = 3x^3 + 10x^2 - x - 12?

Back

No, it is not a factor because f(3) ≠ 0.

Front

Evaluate f(4) for f(x) = x^3 + x^2 - 16x - 16.

Back

f(4) = (4)^3 + (4)^2 - 16(4) - 16 = 64 + 16 - 64 - 16 = 0.

Front

Is (x - 4) a factor of f(x) = x^3 + x^2 - 16x - 16?

Back

Yes, it is a factor because f(4) = 0.

Front

What are the zeros of f(x) = 2x^3 + 5x^2 - 6x - 9 if f(-3) = 0?

Back

The zeros are -1, -3, and 3/2.

Front

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

Back

No, it is not a factor because f(7) ≠ 0.

Front

What is the proper way to write the polynomial 2x^3 - x^2 - 25x - 12?

Back

The proper way is 2x^3 - x^2 - 25x - 12.