Factorisation of Polynomials

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A polynomial in one variable x is an algebraic expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀, where aₙ ≠ 0. Here, a₀, a₁, a₂, ..., aₙ are constants called coefficients, and n i

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Based on number of terms: • Monomial: One term (e.g., 5x³, 7, -2x) • Binomial: Two terms (e.g., x² + 3, 2x - 5) • Trinomial: Three terms (e.g., x² + 2x + 1, 3x³ - x + 4)

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A polynomial of degree one is called a linear polynomial. The general form is ax + b, where a ≠ 0. Examples: 4x + 3, 2x + 5, 7x - 2

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A polynomial of degree two is called a quadratic polynomial. The general form is ax² + bx + c, where a ≠ 0. Examples: x² + 2x + 7, 4x² - x + 1

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A polynomial of degree three is called a cubic polynomial. The general form is ax³ + bx² + cx + d, where a ≠ 0. Examples: x³ + 6x² - 5x + 2, 4x³ - x - 1

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If a polynomial f(x) is divided by a non-zero polynomial g(x), then there exist unique polynomials q(x) and r(x) such that: f(x) = g(x)q(x) + r(x) where either r(x) = 0 or deg r(x) < deg g(x). Here: f

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A non-zero polynomial g(x) is called a factor of polynomial f(x) if and only if there exists a polynomial q(x) such that f(x) = q(x) · g(x), i.e., when f(x) is divided by g(x), the remainder is zero.

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f(3) = (3)² + 5(3) + 4 = 9 + 15 + 4 = 28 Therefore, f(3) = 28