★ Topic Review — Polynomials

Mixed practice covering polynomial basics, division, the remainder and factor theorems, and graphing.

1. Fluency

   For P(x) = 2x⁴ − x³ + 3x − 5:

   • (a) State the degree and leading coefficient.
   • (b) Calculate P(1) and P(−1).
   • (c) Is (0, −5) on the graph? Justify.

2. Fluency

   • (a) Simplify: (3x³ − 2x² + x) − (x³ + x² − 4x + 2).
   • (b) Expand: (x − 2)(x² + x − 3).

3. Fluency

   Use the remainder theorem to find the remainder when each polynomial is divided by the given divisor.

   • (a) P(x) = x³ + 2x² − x + 4, divisor (x − 2)
   • (b) P(x) = 3x³ − x + 5, divisor (x + 1)

4. Fluency

   Use the factor theorem to decide: is (x − 2) a factor of P(x) = x³ − 3x² + 4? Show your working.

5. Fluency

   Perform polynomial long division: (x³ + 4x² − 3x − 18) ÷ (x + 3). Express your answer in the form P(x) = D(x) × Q(x) + R.

6. Understanding

   Fully factorise each polynomial.

   • (a) P(x) = x³ + x² − x − 1
   • (b) P(x) = x³ − 7x + 6

7. Understanding

   For y = −(x + 1)(x − 2)²:

   • (a) State all x-intercepts and their multiplicities.
   • (b) Find the y-intercept.
   • (c) Describe the end behaviour.
   • (d) Describe the graph’s behaviour at x = 2.

8. Understanding

   P(x) = x³ + kx² − 3x + 2 has a remainder of 6 when divided by (x − 2).

   • (a) Use the remainder theorem to write an equation involving k.
   • (b) Solve for k.

9. Understanding

   A quartic polynomial P(x) = x⁴ − 2x³ − 3x² + 4x + 4.

   • (a) Test x = −1 and x = 2 using the factor theorem.
   • (b) Given that −1 and 2 are both double zeros, write P(x) in fully factorised form.
   • (c) Describe the graph’s behaviour at each zero and state the end behaviour.

10. Problem Solving

    P(x) = x³ + ax² − bx − 6. Both (x − 1) and (x + 2) are factors of P(x).

    • (a) Use the factor theorem to write two equations in a and b.
    • (b) Solve the simultaneous equations to find a and b.
    • (c) Write P(x) in fully factorised form.

11. Problem Solving

    A cubic polynomial has x-intercepts at x = −3 and x = 1 (double zero). The graph passes through (2, 5).

    • (a) Write P(x) = a(x − p)(x − q)² using the given zeros.
    • (b) Find the value of a using the given point.
    • (c) Write the complete polynomial in factored form.
    • (d) Find the y-intercept.

12. Problem Solving

    P(x) = x³ + ax² + bx + 6. It is known that (x + 1) is a factor of P(x), and that P(5) = 36.

    • (a) Use the factor condition P(−1) = 0 and the condition P(5) = 36 to write two equations in a and b.
    • (b) Solve simultaneously to find a and b.
    • (c) Fully factorise P(x).
    • (d) Sketch the graph, labelling all intercepts and stating the end behaviour.