📝 Chapter 2: Polynomial Equations & Inequalities

Use synthetic division to divide f(x) = 3x³ - 2x² + 4x - 1 by (x - 2). State the quotient and remainder.

State the Remainder Theorem and the Factor Theorem in your own words.

Factor completely: a) 2x³ + x² - 8x - 4 b) 27x³ + 8

Solve: x³ + x² - 14x - 24 = 0

Determine whether (x + 3) is a factor of 2x⁴ + 5x³ - x² - 6x + 9. Show your work.

Solve the inequality: (x + 1)(x - 2)²(x - 5) ≤ 0. Express your answer in interval notation.

A cubic polynomial f(x) has roots at x = -1, x = 2, and x = 5. Given that f(3) = -16, determine the equation of f(x).

Prove that if f(a) = 0, then (x - a) is a factor of f(x). Use an argument based on the division algorithm.

Without factoring, explain how you can determine the number of positive real roots of f(x) = x⁴ - 3x³ + 2x² + x - 1 using Descartes' Rule of Signs.

A student claims that x⁴ + 1 cannot be factored over the reals. Is this correct? Investigate and explain.

Explain step-by-step how to solve a polynomial inequality using a sign chart / interval test method. Use (x-1)(x+3)(x-4) > 0 as your example.

A classmate asks: "What's the difference between a root, a zero, an x-intercept, and a factor?" Write a clear explanation connecting all four concepts.

Create a concept map or flowchart showing the steps for factoring a polynomial of degree 3 or higher. Include: testing potential rational roots, synthetic division, and recognizing special forms.

Write a journal entry (5-8 sentences) reflecting on: Which factoring technique (grouping, rational root theorem, synthetic division, or special forms) do you find most challenging and why?

A company's profit P(x) = -x³ + 12x² - 36x, where x is items produced in thousands. a) For what production levels is profit positive? b) What production level maximizes profit? (Use graphing/calculus concepts)

The dimensions of a box are (x+1), (x-2), and (x+3) cm. a) Write the volume as a polynomial. b) If the volume is 120 cm³, find x. c) State the actual dimensions.

The path of a projectile is modeled by h(t) = -5t³ + 20t² + 15t where h is height in metres and t is time in seconds. a) When does the projectile hit the ground? b) During what time interval is the projectile above 30m?

Design a polynomial function with ALL of the following properties: degree 5, exactly 3 real roots, y-intercept of 12, and end behaviour from Q2 to Q4. Write its equation and sketch its graph.