Polynomials

Overview

Polynomials form a foundational topic in upper-primary mathematics, bridging arithmetic and advanced algebra. For KAR TET Paper II, you must understand what polynomials are, how to find their zeros, and how to apply the division algorithm—skills essential for teaching Classes 6–8 students. This topic directly connects to factorisation, quadratic equations, and algebraic identities in the syllabus.

Expect questions testing your ability to identify polynomial types, find zeros graphically and algebraically, verify relationships between zeros and coefficients, and perform polynomial division. Mastery here also supports pedagogy questions on teaching algebraic thinking to young learners.

Key Concepts

• **Polynomial definition**: An algebraic expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where coefficients are real numbers and exponents are whole numbers (non-negative integers).

• **Degree of a polynomial**: The highest power of the variable with a non-zero coefficient. Example: 3x⁴ + 2x – 7 has degree 4.

• **Types by degree**: Constant (degree 0), Linear (degree 1), Quadratic (degree 2), Cubic (degree 3), Biquadratic/Quartic (degree 4).

• **Zero/Root of a polynomial**: A value 'a' such that p(a) = 0. Geometrically, zeros are the x-coordinates where the graph crosses or touches the x-axis.

• **Number of zeros**: A polynomial of degree n has at most n real zeros. A linear polynomial has exactly 1 zero; a quadratic has at most 2; a cubic has at most 3.

• **Relationship between zeros and coefficients**: For quadratic ax² + bx + c with zeros α and β: Sum (α + β) = –b/a; Product (αβ) = c/a.

• **Division Algorithm**: For polynomials p(x) and g(x) where g(x) ≠ 0, there exist unique q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where degree of r(x) < degree of g(x) or r(x) = 0.

• **Factor Theorem**: (x – a) is a factor of p(x) if and only if p(a) = 0.

Formulas / Key Facts

| Formula/Fact | Context | |--------------|---------| | p(a) = 0 implies 'a' is a zero | Definition of zero/root | | Linear p(x) = ax + b has zero at x = –b/a | Finding zero of linear polynomial | | Quadratic: α + β = –b/a | Sum of zeros for ax² + bx + c | | Quadratic: αβ = c/a | Product of zeros for ax² + bx + c | | Cubic: α + β + γ = –b/a | Sum of zeros for ax³ + bx² + cx + d | | Cubic: αβ + βγ + γα = c/a | Sum of products taken two at a time | | Cubic: αβγ = –d/a | Product of all three zeros | | Division Algorithm: Dividend = Divisor × Quotient + Remainder | Always applies; remainder degree < divisor degree | | Maximum real zeros = Degree of polynomial | Upper bound on number of zeros |