Polynomials and the factor theorem · Grade 12
In Grade 12 you learn: dividing polynomials.
A polynomial is an expression made from whole-number powers of x, like x³ − 4x² + x + 6. The factor and remainder theorems let you find factors, roots and remainders of polynomials quickly, without dividing every time.
Key facts
- The degree is the highest power of x: 2x³ − x + 5 has degree 3.
- Remainder theorem: dividing p(x) by (x − a) leaves remainder p(a).
- Factor theorem: (x − a) is a factor of p(x) exactly when p(a) = 0.
- Dividing by (ax − b) leaves remainder p(b/a).
- To solve a cubic: find one root by trying factors of the constant term, divide by that factor, then factorise the quadratic left over.
- A repeated factor (x − a)² means the graph touches the x-axis at x = a without crossing.
Worked examples
Example 1
Find the remainder when x³ − 3x² + 5x − 4 is divided by (x − 2).
Answer: 2
Example 2
Show that (x + 1) is a factor of x³ − 4x² + x + 6, and solve x³ − 4x² + x + 6 = 0.
Answer: x = −1, 2 or 3