Factor Theorem

If p(x)=(x-a)(x-b)…=0, then either x=a or x=b…

So each linear factor shows us a root (and conversely).

Factor Theorem: p(t)=0 <=> x-t is a factor of p(x)

When polynomials have small coefficients this can help us quickly find factors. We need only consider the divisors of the constant as any integer factor must divide it.

Extended Factor Theorem: p(t/s)=0 <=> sx-t is a factor of p(x).

Worked Examples

• 1.) Given that f(x) = x³ – 6x² + 11x – 6,
  • Find f(0), f(1), f(2), f(3) and f(4);
  • Factorise x³ – 6x² + 11x – 6;
  • Solve the equation x³ – 6x² + 11x – 6 = 0;
  • Sketch the curve whose equation is f(x) = x³ – 6x² + 11x – 6
• 2.) Given that f(x) = x³ – x² – 3x + 2,
  • Show that x-2 is a factor;
  • Solve the equation f(x) = 0.

Exercise

Solutions