Polynomials Notes

A polynomial in x is an expression of the form aₙxⁿ + … + a₁x + a₀ where the coefficients are real numbers and n is a non-negative integer. Expressions with negative or fractional powers (like x⁻¹ or x^(1/2)) are not polynomials.

The degree is the highest power of the variable with a non-zero coefficient. A polynomial of degree 1 is linear, degree 2 is quadratic, degree 3 is cubic. A non-zero constant has degree 0; the zero polynomial has no defined degree.

The value of a polynomial p(x) at x = a is p(a), obtained by substituting a for x.

A real number a is a zero (or root) of p(x) if p(a) = 0. Finding zeroes means solving p(x) = 0.

A linear polynomial ax + b has exactly one zero, x = −b/a. A polynomial of degree n has at most n real zeroes. A non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.

Example: for p(x) = x² − 5x + 6, p(2) = 4 − 10 + 6 = 0 and p(3) = 9 − 15 + 6 = 0, so 2 and 3 are the zeroes.

Remainder Theorem: if a polynomial p(x) is divided by the linear polynomial (x − a), the remainder is p(a). So you can find the remainder without doing the division.

Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0. More generally, (x − a) is a factor exactly when a is a zero of p(x).

Example: is (x − 1) a factor of p(x) = x³ − 3x² + 3x − 1? p(1) = 1 − 3 + 3 − 1 = 0, so yes. In fact p(x) = (x − 1)³.

Identities let you expand and factorise quickly without multiplying out every time. The most used ones in Class 9 are the squares of a binomial, the difference of two squares, the product (x + a)(x + b), and the cubes.

For three variables, (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx is common, and the identity x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx) is used in special factorisations (note that if x + y + z = 0 then x³ + y³ + z³ = 3xyz).

To factorise a quadratic ax² + bx + c, split the middle term: find two numbers whose product is ac and whose sum is b.