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Aptitude Topics

Polynomials

A polynomial is an expression with one or more terms involving variables raised to non-negative integer powers. Mastering polynomials is essential for factoring, equation-solving, and algebraic manipulation.

Fundamental Principles

Polynomial

An expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ where the exponents are non-negative integers and coefficients are real numbers.

Degree of a Polynomial

The highest power of the variable in the polynomial. E.g., 3x⁴ - 2x + 5 has degree 4.

Factor Theorem

A polynomial p(x) has (x - a) as a factor if and only if p(a) = 0.

Remainder Theorem

When a polynomial p(x) is divided by (x - a), the remainder is p(a).

Essential Formulation Tips

  • Use the remainder theorem before doing full polynomial division.
  • Check if (x ± simple integers) are factors by substituting them into p(x).
  • Long division and synthetic division yield the same result — choose what's faster.
  • Group terms strategically for factoring by grouping.

Shortcut Execution Techniques

  • p(a) = 0 → (x - a) is a factor.
  • Sum of all coefficients = p(1). If p(1) = 0, then (x-1) is a factor.
  • Sum of coefficients of even terms = sum of odd terms → (x+1) is a factor.
  • Degree of product = sum of degrees.

Contextual Inquiries (FAQs)

Q: What is the remainder when p(x) is divided by (x - 2)?

A: The remainder is p(2) by the Remainder Theorem.

Q: How do you verify if (x + 3) is a factor?

A: Check if p(-3) = 0. If yes, (x + 3) is a factor.

Q: What is a zero of a polynomial?

A: A value of x for which p(x) = 0. Zeros are the same as roots.