Grade 12 Cubic Equations Worksheet
A comprehensive worksheet for Grade 12 students covering cubic equations, including solving, factoring, and real-world applications.
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Grade 12 Cubic Equations Worksheet
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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?
-1
0
1
2
2. If (x - 2) is a factor of P(x) = x³ - 3x² + kx - 10, what is the value of k?
1
2
3
4
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.
True
False
9. If a cubic equation has a complex root (a + bi), then its conjugate (a - bi) must also be a root.
True
False
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.