• Quick retrieval from earlier learning (Remainder and factor theorems). Students answer on mini whiteboards.
• Show answers together and note who needs support in the main task.

1. Find the remainder when x³ − 2x² + 5x + 7 is divided by (x + 2). → -19Prompt: Remainder theorem: the remainder on dividing by (x − a) is f(a).
2. (x − 1) is a factor of x³ − 7x² + 14x − 8. Solve x³ − 7x² + 14x − 8 = 0. Give all three solutions, separated by commas. → x = 1 or x = 2 or x = 4Prompt: Divide by the known factor to get a quadratic.
3. Find the remainder when x³ + x² − 8x + 7 is divided by (x + 2). → 19Prompt: Remainder theorem: the remainder on dividing by (x − a) is f(a).

• Show a "think of a number" puzzle and ask the class to work out the starting number.
• Share the objective: Polynomial functions: zeros and end behaviour. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: f(x) = −(x − 2)(x − 3)(x + 3). Find all the zeros of f, separated by commas.
• Step 1: x = −3, 2, 3
• Answer: x = -3 or x = 2 or x = 3
• Pause after each step and ask a student to say what comes next.

• Do these together. Students try each step on whiteboards before you reveal it.
• Use the hints as prompts rather than giving the answer.

1. f(x) = 2(x)(x + 6)(x − 3). Find all the zeros of f, separated by commas. → x = -6 or x = 0 or x = 3Prompt: A product is zero when one of its factors is zero.
2. Describe the end behaviour of f(x) = 2x⁴ + 3x − 1. (Up on the left, up on the right / Down on the left, down on the right / Down on the left, up on the right / Up on the left, down on the right) → Up on the left, up on the rightPrompt: Only the leading term matters for very large |x|.

• Students work on the student sheet. Circulate and check the first two answers of every student early.
• Students who finish: write a question of their own for a partner, with the answer.

The 8 questions are on the student sheet below, with answers in the key.

• Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.

1. f(x) = 2(x − 5)(x − 3)(x − 1). Find all the zeros of f, separated by commas. → x = 1 or x = 3 or x = 5Prompt: A product is zero when one of its factors is zero.
2. Describe the end behaviour of f(x) = 2x⁵ + 3x − 1. (Up on the left, up on the right / Down on the left, down on the right / Down on the left, up on the right / Up on the left, down on the right) → Down on the left, up on the rightPrompt: Only the leading term matters for very large |x|.

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.