All subjects · Mathematics · WMA11 (P1)

Polynomials and cubic graphs

A cubic has x3 as its highest power. If you can factorise it, you can sketch it quickly from its roots.

Expanding and factorising

Expand two brackets first, then multiply by the third, and collect like terms. To factorise a cubic with no constant term, take out x first: x3 - 4x = x(x2 - 4) = x(x - 2)(x + 2).

Sketching y = a(x - p)(x - q)(x - r)

Repeated roots

A squared factor such as (x - 2)2 gives a repeated root: the curve touches the x-axis there and turns, instead of crossing. y = x3 has a triple root at 0 and flattens out as it passes through the origin.

Worked example

Sketch y = x3 - 4x: find its roots and y-intercept.

  1. Factorise: x(x2 - 4) = x(x - 2)(x + 2)
  2. Roots: x = -2, 0, 2
  3. x = 0 gives y = 0
  4. Positive x3 coefficient: bottom left to top right

Answer: Crosses the x-axis at -2, 0 and 2 and passes through the origin.

Key idea: Factorise, mark the roots and the y-intercept, and use the sign of the x3 term for the shape. A squared factor touches the axis.

Check yourself

  1. Expand (x + 2)(x - 1)(x + 3).
    Show answer

    x3 + 4x2 + x - 6. (x + 2)(x - 1) = x2 + x - 2. Multiply by (x + 3): x3 + 3x2 + x2 + 3x - 2x - 6 = x3 + 4x2 + x - 6.

  2. Solve x3 - 9x = 0.
    Show answer

    x = 0, x = 3 or x = -3. x(x2 - 9) = x(x - 3)(x + 3) = 0.

  3. Where does y = (x - 1)(x + 2)(x - 4) cross the y-axis? Give the y-coordinate.
    Show answer

    8. Put x = 0: (-1)(2)(-4) = 8.

  4. For y = (x - 2)2(x + 1), what happens at x = 2?
    • The curve crosses the x-axis
    • The curve has an asymptote
    • The curve touches the x-axis and turns
    • The curve crosses the y-axis
    Show answer

    The curve touches the x-axis and turns. (x - 2)2 is a repeated root, so the curve touches the x-axis there.

  5. For y = -(x + 1)(x - 2)(x - 3), what happens to y as x becomes large and positive?
    • y becomes large and negative
    • y becomes large and positive
    • y gets closer to 0
    • y gets closer to 3
    Show answer

    y becomes large and negative. The x3 coefficient is -1, so for large positive x, y is large and negative.

  6. Solve x3 - 5x2 + 6x = 0.
    Show answer

    x = 0, x = 2 or x = 3. x(x2 - 5x + 6) = x(x - 2)(x - 3) = 0.

Practise with instant feedback

Sources

Last updated 7 November 2026. Spotted a mistake? Tell us.