• 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 + 11x + 30 = 0. Give both solutions, separated by a comma. → x = -6 or x = -5Prompt: Factorise: find two numbers that multiply to 30 and add to 11.
2. Divide x³ + 9x² + 20x by (x + 4). The remainder is 0. Type the quotient. → x^2+5xPrompt: Use long division or synthetic division with −4.
3. Divide x³ + 5x² + 6x + 3 by (x + 3). Type the quotient (ignore the remainder). → x^2+2xPrompt: Bring down the leading coefficient, multiply by −3, 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 = 2 is a solution of x³ − 2x² − 16x + 32 = 0. Find all the solutions, separated by commas. → x = -4 or x = 2 or x = 4Prompt: So (x − 2) is a factor: divide it out.
2. Solve x³ + 7x² + 12x = 0. Give all the solutions, separated by commas. → x = -4 or x = -3 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 6 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 = 4 is a solution of x³ + 3x² − 18x − 40 = 0. Find all the solutions, separated by commas. → x = -5 or x = -2 or x = 4Prompt: So (x − 4) is a factor: divide it out.
2. Solve x³ + 5x² − 6x = 0. Give all the solutions, separated by commas. → x = -6 or x = 0 or x = 1Prompt: 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.