Polynomial functions; factor and remainder theorems practice questions.

Calculate the remainder when P(x) = x³ - 2x² + 3x - 5 is divided by (x - 2).

• -27
• 1
• -5
• 9

Answer: 1

By the remainder theorem the remainder is P(2) = 8 - 8 + 6 - 5 = 1. Evaluating P(-2) instead is the classic sign error, giving -27.

Given that (x + 3) is a factor of P(x) = x³ + kx² - 4x - 12, determine the value of k.

• 3
• -3
• -1/3
• 9

Answer: 3

By the factor theorem P(-3) = 0: -27 + 9k + 12 - 12 = 0 gives 9k = 27, so k = 3. Indeed x³ + 3x² - 4x - 12 = (x + 3)(x² - 4).

Determine the sum of the roots of the equation 2x³ - 5x² + x - 7 = 0.

• -5/2
• 7/2
• -1/2
• 5/2

Answer: 5/2

For ax³ + bx² + cx + d = 0 the sum of the roots is -b/a = -(-5)/2 = 5/2. Forgetting the minus sign in the formula gives the distractor -5/2.

Determine the product of the roots of the equation x⁴ - 3x³ + 2x - 8 = 0.

• 8
• -2
• -8
• 2

Answer: -8

For a degree-n polynomial the product of the roots is (-1)ⁿ · (constant term)/(leading coefficient) = (-1)⁴ · (-8)/1 = -8.

Calculate the remainder when P(x) = 4x³ + 2x² - 3x + 1 is divided by (2x - 1).

• 1/2
• 5/2
• 0
• 1

Answer: 1/2

Dividing by (2x - 1) gives remainder P(1/2) = 4/8 + 2/4 - 3/2 + 1 = 1/2. Using P(2) or P(-1/2) are the standard mistakes.

A cubic polynomial P(x) has a double root at x = 1, a single root at x = -2, and leading coefficient 3. Calculate P(0).

• -6
• 2
• 6
• 3

Answer: 6

P(x) = 3(x - 1)²(x + 2), so P(0) = 3 × (-1)² × 2 = 6. The double root contributes a squared factor, which removes its negative sign at x = 0.