Revision: Polynomials Algebra SSC (English Medium) 9th Standard Maharashtra State Board

When an algebraic expression is made of only one variable, it is called a polynomial in one variable.

Examples of Polynomials in One Variable:

A polynomial is an algebraic expression made up of terms in which the variables have non‑negative whole-number exponents.

The degree of a polynomial is simply the highest exponent (power) in the expression.

Example 1: 4x² - 3x⁵ + 8x⁶

• Term 1: 4x² → exponent = 2
• Term 2: -3x⁵ → exponent = 5
• Term 3: 8x⁶ → exponent = 6
• Degree = 6 (highest exponent)

Example 2: 25 - x⁴

• Term 1: 25 → exponent = 0 (since 25 = 25x⁰)
• Term 2: -x⁴ → exponent = 4
• Degree = 4

The highest power of the variable in a polynomial is called its degree.

An expression of the form

f(x) = a_₀xⁿ + a_₁xⁿ⁻¹ + a_₂xⁿ⁻² + … + aₙ₋₁x + aₙ,

where a_₀, a_₁, a_₂, …, aₙ₋₁, aₙ are real numbers and a_₀ ≠ 0, is called a polynomial of degree n

• Degree of a polynomial = highest power of the variable.

• Leading term: term with the highest power.

• Leading coefficient: coefficient of highest power.

• Constant term: term without the variable.

A function f(x) is a rule or expression whose value depends on the variable x.

The value of the function at x = a is denoted by f(a) and is obtained by substituting x = a in f(x).

A real number k is a zero of p(x) if p(k) = 0.

A polynomial g(x) is called a factor of the polynomial f(x) if g(x) divides f(x) exactly, giving 0 as the remainder.

For

p(x) = ax + b

Zero: \[x=-\frac{b}{a}\]​

Quadratic polynomial

$,\; a\; \ne \; 0$

Cubic polynomial

$,\; a\; \ne \; 0$

Statement:
If a polynomial f(x) is divided by (x − a), then the remainder is f(a).

Result:
$.$

Statement

If a polynomial f(x) is divided by (x − a) and the remainder is zero, then (x − a) is a factor of f(x).

Result

(x − a) is a factor of f(x)  ⟺  f(a) = 0

To check whether (x − a) is a factor → find f(a)

• If f(a) = 0 → factor

• If f(a) ≠ 0 → not a factor

Important Forms

• (x − a) is a factor ⇔ f(a) = 0

• (x + a) is a factor ⇔ f(−a) = 0

• (ax + b) is a factor ⇔ \[f(-\frac{b}{a})\] = 0