• Quick retrieval from earlier learning (Divide a polynomial by (x − a); Solve quadratics by factorising). Students answer on mini whiteboards.
• Show answers together and note who needs support in the main task.

1. Solve: x² + 13x + 40 = 0. Give both solutions, separated by a comma. → x = -8 or x = -5Prompt: Factorise: find two numbers that multiply to 40 and add to 13.
2. Divide x³ + 2x² − 11x − 12 by (x − 3). The remainder is 0. Type the quotient. → x^2+5x+4Prompt: Use long division or synthetic division with 3.
3. Divide x³ − 3x² − 13x − 8 by (x + 2). Type the quotient (ignore the remainder). → x^2-5x-3Prompt: Bring down the leading coefficient, multiply by −2, add, and repeat.

• Show a "think of a number" puzzle and ask the class to work out the starting number.
• Share the objective: Solve a cubic equation when one root is known. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: x = −5 is a solution of x³ + 3x² − 13x − 15 = 0. Find all the solutions, separated by commas.
• Step 1: (x + 5)(x² − 2x − 3) = 0
• Step 2: (x + 5)(x − 3)(x + 1) = 0
• Step 3: x = −5, −1, 3
• Answer: x = -5 or x = -1 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. x = 3 is a solution of x³ − 6x² + 11x − 6 = 0. Find all the solutions, separated by commas. → x = 1 or x = 2 or x = 3Prompt: So (x − 3) is a factor: divide it out.
2. Solve x³ + 7x² + 10x = 0. Give all the solutions, separated by commas. → x = -5 or x = -2 or x = 0Prompt: Every term has a factor of x: take it out.

• 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. x = −3 is a solution of x³ + x² − 14x − 24 = 0. Find all the solutions, separated by commas. → x = -3 or x = -2 or x = 4Prompt: So (x + 3) is a factor: divide it out.
2. Solve x³ + 2x² − 15x = 0. Give all the solutions, separated by commas. → x = -5 or x = 0 or x = 3Prompt: Every term has a factor of x: take it out.

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-11.