Remainder theorem, factor theorem and roots of polynomials
Find a remainder without dividing, spot a factor with the factor theorem, find the roots of a cubic and read coefficients from roots.
- UK: Year 12
- Germany: grade 10
- Iran: grade 12
- Afghanistan: grade 10
- Russia: grade 10
Guide
remainder theorem: when a polynomial p(x) is divided by x − a, the remainder is p(a). Do not divide; just put the number a into the polynomial. a is where x − a is 0, for example −2 for x + 2.
factor theorem: if p(a) = 0 the remainder is 0 and x − a is a factor of p(x). So a is a root of p(x) = 0.
To solve a cubic, first find one root: try small whole numbers that divide the constant term. Take out that factor; a quadratic is left, and it factorises.
If you know the roots of x3 + px2 + qx + r, expand (x − a)(x − b)(x − c) and read the coefficients. In each task type the number or mark the roots as points on the number line.