We will discuss here about the basic concept of factors of a polynomial.
We have, f(x) = ϕ(x) ∙ ψ(x) + R(x), where R(x) is the remainder and ψ(x) is the quotient when f(x) is divided by ϕ(x).
If R(x) = 0, f(x) is divided by ϕ(x) and f(x) = ϕ(x) ∙ ψ(x).
ϕ(x) and ψ(x) are factors of f(x).
Examples on factors of a polynomial:
(i) If x2 - x - 12 is divided by x - 4 then
Therefore, the remainder = 0, and x^2 - x - 12 = (x - 4)(x + 3).
Therefore, (x - 4) and (x + 3) are factors of the quadratic polynomial x^2 - x - 12.
(ii) If x^3 + 2x^2 + x + 2 is divided by x + 2 then
Therefore, the remainder = 0, and x^3 + 2x^2 + x + 2 = (x + 2)(x^2 + 1).
Therefore, (x + 2) and (x^2 + 1) are factors of the cubic polynomial x^3 + 2x^2 + x + 2.
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