Ch 2 formalises polynomial terminology, introduces the Remainder and Factor Theorems, and extends algebraic identities. Students learn to find zeroes of polynomials and factorise cubic expressions.
A polynomial in x is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀. The degree is the highest power with non-zero coefficient. A zero of p(x) is a value c such that p(c) = 0.
Remainder Theorem: if p(x) is divided by (x − a), the remainder is p(a). Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0. These are powerful tools for factorisation.
(a+b+c)² = a²+b²+c²+2ab+2bc+2ca. (a+b)³ = a³+3a²b+3ab²+b³. (a−b)³ = a³−3a²b+3ab²−b³. a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca). If a+b+c=0, then a³+b³+c³ = 3abc.
Download: https://ncert.nic.in/textbook/pdf/iemh102.pdf | Complete book: https://ncert.nic.in/textbook/pdf/iemh1ps.zip
The Factor Theorem states that (x − a) is a factor of polynomial p(x) if and only if p(a) = 0. This links the algebraic concept of factors with the numerical concept of zeroes.
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