Polynomials Introduction Chapter 2 Class 9 SHIKHAR 2024 BYJUS

Polynomials: Degree, Zeros, and the Factor Theorem

A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, with non-negative integer exponents on the variables. The word "polynomial" comes from "poly" (many) and "nomial" (term). In CBSE Class 9 Mathematics, Chapter 2 introduces students to the classification of polynomials by degree and number of terms, the relationship between zeros and coefficients, and the powerful Remainder and Factor Theorems that simplify polynomial division and factorisation.

Polynomials are classified by degree — the highest power of the variable. A constant polynomial (degree 0) is simply a number, like 5. A linear polynomial (degree 1) has the form ax + b, where a ≠ 0. A quadratic polynomial (degree 2) has the form ax² + bx + c. A cubic polynomial (degree 3) has the form ax³ + bx² + cx + d. The degree of a polynomial in one variable determines the maximum number of zeros it can have — a polynomial of degree n has at most n zeros. A zero (or root) of a polynomial p(x) is a number α such that p(α) = 0. Geometrically, the zeros of a polynomial correspond to the x-intercepts of its graph — the points where the curve crosses the x-axis. A linear polynomial has exactly one zero (x = −b/a). A quadratic polynomial can have two distinct zeros, one repeated zero (the vertex touches the x-axis), or no real zero (the parabola does not cross the x-axis).

The Remainder Theorem provides a shortcut for finding the remainder when a polynomial p(x) is divided by a linear polynomial (x − a): the remainder is simply p(a). For example, to find the remainder when x³ − 2x + 1 is divided by (x − 2), just evaluate p(2) = 8 − 4 + 1 = 5. The Factor Theorem is a special case of the Remainder Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0. This is extremely useful because it provides a quick test for whether a given linear expression divides a polynomial evenly. To factorise a cubic or higher-degree polynomial, first find one zero by testing simple values (±1, ±2, ±3), then divide by (x − a) using long division or synthetic division to get a quotient of lower degree, which can then be factorised further. The algebraic identities help in factorisation: x² − y² = (x + y)(x − y), x² + 2xy + y² = (x + y)², x³ + y³ = (x + y)(x² − xy + y²), and x³ − y³ = (x − y)(x² + xy + y²). For quadratic polynomials ax² + bx + c, the relationship between zeros and coefficients is: sum of zeros = −b/a and product of zeros = c/a. These relationships allow students to find zeros given the polynomial or construct a polynomial given its zeros.

• Polynomials are classified by degree: constant (0), linear (1), quadratic (2), cubic (3); degree n means at most n zeros.
• A zero of p(x) is a value α where p(α) = 0; geometrically these are the x-intercepts of the graph.
• Remainder Theorem: dividing p(x) by (x − a) gives remainder p(a); Factor Theorem: (x − a) is a factor when p(a) = 0.
• Factorise by finding one zero, dividing, then factoring the quotient; use algebraic identities for special patterns.
• For quadratic ax² + bx + c: sum of zeros = −b/a, product of zeros = c/a.