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Polynomials
Chapter summary, hard words and model exam answers for ICSE Class 9 Hindi.
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Maths · ICSE Class 9
Summary
A polynomial in x is made using constants, x, and non-negative whole-number powers of x. Expressions like 4x^3 - 2x + 7 are polynomials. Expressions with x in the denominator or fractional powers of x are not polynomials in x.
The degree is the highest power of x whose coefficient is not zero. Degree 0 is constant, degree 1 is linear, degree 2 is quadratic and degree 3 is cubic. The zero polynomial has no defined degree.
To find p(a), replace x by a and simplify. If p(a) = 0, then a is a zero of the polynomial. The word zero here means an input that makes the polynomial value zero.
When p(x) is divided by x - a, the remainder is p(a). If p(a) = 0, then x - a is a factor of p(x). This is a fast way to test factors without doing full division.
An identity is true for every allowed value. Use identities to expand and factor quickly, but only after matching the structure. For example, (a + b)(a - b) is a difference of squares.
Factorisation writes a polynomial as a product. First look for a common factor, then try identities, splitting the middle term, grouping, or the factor theorem.
Hard words & meanings
| term | One part of an algebraic expression separated by plus or minus signs |
| coefficient | The numerical factor multiplying a variable term |
| degree | The highest power of the variable in a non-zero polynomial |
| zero of a polynomial | A value of the variable that makes the polynomial equal to zero |
| remainder theorem | For division by x - a, the remainder is p(a) |
| factor theorem | If p(a) = 0, then x - a is a factor of p(x) |
| identity | An equation true for all allowed values of its variables |
Model exam answers, grammar & audio
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