2 · IB Maths AA
Polynomials and the factor theorem
Drag the gold roots along the x-axis (or use the sliders) to build a cubic or quartic, then move the red test value k to see the factor theorem, the remainder, and how the sum and product of the roots come straight from the coefficients.
Key equations
p(x) = a(x − r₁)(x − r₂)…(x − rₙ)
Factor theorem: p(k) = 0 ⇔ (x − k) is a factor
Remainder theorem: p(x) = (x − k)q(x) + p(k)
Cubic ax³ + bx² + cx + d: Σr = −b/a, product = −d/a
Quartic ax⁴ + bx³ + cx² + dx + e: Σr = −b/a, product = e/a
Try this
- Drag two roots onto the same value: the graph now touches the axis instead of crossing it, and the factored form shows a squared bracket. Check that the sum and product still match.
- Keep the roots fixed and change a from 1 to −2: every coefficient scales, the graph flips, but −b/a and the product formula give the same roots as before.
- Slide k onto one of the roots and watch the remainder become 0; the quotient q(x) is then exactly the product of the other factors.
Exam tip: when using the factor theorem, write the substitution p(k) = 0 out in full before concluding that (x − k) is a factor, then divide to find the remaining quadratic.