Horner's scheme and integer roots
Divide a polynomial by x − a with Horner's table, find the remainder and the integer roots among the divisors of the constant term.
- Russia: grade 10
Guide
In the Russian programme, dividing polynomials, the remainder (Bezout) theorem and integer roots are studied in grade 10 (algebra, advanced level; not required in the basic course). Horner's scheme is a handy table for dividing a polynomial by x − a.
The top row holds the coefficients from the highest power down (a missing power gets a 0). In the bottom row bring down the first coefficient; each next number is a · (number on the left) + (coefficient above). The last number is the remainder, equal to P(a) (remainder theorem); the others are the coefficients of the quotient.
An integer root of a polynomial with integer coefficients and leading coefficient 1 divides the constant term. Try the divisors, find a root, divide with Horner's scheme and solve the equation of lower degree.