Revision: Algebra >> Remainder Theorem and Factor Theorem Maths (English Medium) ICSE Class 10 CISCE

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).

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

Statement:
On dividing a polynomial f(x)by a polynomial g(x), there exist polynomials q(x) and r(x) such that

f(x) = g(x)q(x) + r(x)

where either r(x) = 0 or degree of r(x) < degree of g(x)

Result:
degree of r(x) < degree of g(x)

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