Polynomials
Polynomials tests zeroes, coefficient relationships, the division algorithm, and graphs. RBSE items often give a quadratic and ask for a polynomial with prescribed zeroes or verify relationships.
Core rules
- A polynomial of degree n has at most n real zeroes. A quadratic ax^2+bx+c, a ≠ 0, has sum of zeroes -b/a and product c/a.
- For a cubic ax^3+bx^2+cx+d, sum = -b/a, sum of products taken two at a time = c/a, product = -d/a.
- Division algorithm: p(x) = g(x)q(x) + r(x), where either r(x)=0 or degree r < degree g.
- Remainder theorem: the remainder on dividing p(x) by x-a is p(a). Factor theorem: x-a is a factor exactly when p(a)=0.
- The graph of a quadratic is a parabola; the number of x-intercepts equals the number of distinct real zeroes.
Worked example
Find a quadratic polynomial whose zeroes are 2 and -3. Sum = -1, product = -6. Using x^2 - (sum)x + product gives x^2 + x - 6. Check by factorising: (x-2)(x+3)=x^2+x-6. The coefficient relation also holds: -b/a = -1/1 = -1 and c/a = -6.
Traps
Keep signs with the formulas: sum is -b/a, not b/a. Zeroes are not coefficients. A zero polynomial has no defined degree, so do not call it degree 0. When using the factor theorem, substitute carefully, especially with negative values.