Exploring Remainders and Factors

What does the Factor Theorem state about a polynomial f(x) and a root r?

• If (x - r) is a factor, then f(r) must equal 1.

• If f(r) = 0, then (x - r) is a factor of f(x).

• If f(r) = 0, then (x + r) is a factor of f(x).

• If f(r) ≠ 0, then (x + r) is a factor of f(x).

If f(x) = x^3 - 6x^2 + 11x - 6, which of the following is a root?

• 3

• 4

• 1

• 2

Using the Factor Theorem, if f(2) = 0, what can we conclude about (x - 2)?

• (x - 2) is a factor of f(x).

• (x - 2) is not a factor of f(x).

• f(2) is undefined.

• f(x) has no real roots.

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

• 1

• 3

• 0

• -1

Which of the following is a factor of f(x) = x^2 + 5x + 6?

• (x + 2)(x + 3)

• (x + 1)(x + 6)

• (x + 4)(x + 5)

• (x - 2)(x - 3)

If f(x) = x^3 - 4x + 4, what is f(-2)?

• 8

• 4

• 0

• -4

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

• -4

• -5

• 0

• -3

If r is a root of f(x), what can be said about f(r)?

• f(r) = 0

• f(r) > 0

• f(r) = 1

• f(r) < 0

Which polynomial has a root at x = 1?

• x + 1

• x - 1

• x^2 - 1

• x - 2

If f(x) = x^2 - 5x + 6, what are the roots of the polynomial?

• 1, 4

• 0, 6

• -1, 5

• 2, 3