Grade 12 Cubic Equations Worksheet

Grade 12 Cubic Equations Worksheet

Read each question carefully and show all your work. Use appropriate mathematical notation. Round answers to two decimal places where necessary.

1. Which of the following is a root of the cubic equation x³ - 6x² + 11x - 6 = 0?

2. If (x - 2) is a factor of P(x) = x³ - 3x² + kx - 10, what is the value of k?

3. According to the Rational Root Theorem, possible rational roots of 2x³ + 3x² - 8x + 3 = 0 are  .

4. A cubic equation can have a maximum of   real roots.

5. If a cubic polynomial has roots at x = 1, x = -2, and x = 3, its equation can be written as P(x) = k(x - 1)(x + 2)(x - 3). If P(0) = 12, then k =  .

6. Find all real roots of the cubic equation x³ + 2x² - 5x - 6 = 0.

7. Factor the polynomial P(x) = x³ - 7x + 6 completely.

8. Every cubic equation with real coefficients has at least one real root.

9. If a cubic equation has a complex root (a + bi), then its conjugate (a - bi) must also be a root.

10. Consider the function f(x) = x³ - 3x² + 2x. Sketch the graph of this function on the coordinate plane below, identifying its x-intercepts and y-intercept.

11. A rectangular box has a volume given by the cubic polynomial V(x) = x³ - 7x² + 14x - 8, where x represents the length of one side. If the width is (x-1) and the height is (x-4), find the length of the box.