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10 Mathematics notes Chapter 2 Polynomials

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CBSE Class–10 Mathematics
Revision Notes
CHAPTER 02
POLYNOMIALS

Geometrical Meaning of the Zeroes of a Polynomial
Zeroes and Coefficients of a Polynomial
Division Algorithm for Polynomials
Monomials: Algebraic expression with one term is known as Monomial.
Binomial: Algebraic expression with two terms is called Binomial.
Trinomial: Algebraic expression with three terms is known as Trinomial.
Polynomials: All above mentioned algebraic expressions are called Polynomials.
Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
A quadratic polynomial in x with real coefficient is of the form ${\text{a}}{{\text{x}}^2}{\text{ }} + {\text{ bx }} + {\text{ c}}$ ,where a, b, c are real numbers with ${\text{a }} \ne 0$ .
The zeroes of a polynomial p(x) are precisely the x–coordinates of the points where the graph of y = p(x) intersects the x-axis i.e. x = a is a zero of polynomial p(x) if p(a) = 0.
A polynomial can have at most the same number of zeros as the degree of polynomial.
For quadratic polynomial ${\text{a}}{{\text{x}}^2} + {\text{ bx }} + {\text{ c}}$ $(a \ne 0)$

Sum of zeroes = $\; - \frac{b}{a}$

Product of zeroes = $\;\frac{c}{a}$

10. In a cubic polynomial $a{x^3} + b{x^2} + cx + d$ , if $\alpha ,\beta ,\gamma$ are the zeroes of the polynomial, then

$\alpha + \beta + \gamma = {{ - b} \over a}$

$\alpha \beta + \beta \gamma + \gamma \alpha = {c \over a}$

$\alpha .\beta .\gamma = {d \over a}$

11. The division algorithm states that given any polynomial p(x) and polynomial g(x), there are polynomials q(x) and r(x) such that :

${\text{p}}\left( {\text{x}} \right){\text{ }} = \;{\text{g}}\left( {\text{x}} \right).{\text{q }}\left( {\text{x}} \right){\text{ }} + \;{\text{r}}\left( {\text{x}} \right),\;\;{\text{g}}\left( {\text{x}} \right){\text{ }} \ne 0$

where r(x) = 0 or degree of r(x) < degree of g(x).

Or Dividend = Divisor x Quotient + Remainder

If r(x) = a zero polynomial, then p(x) is said to be completely divisible by g(x), i.e., g(x) is a factor of p(x).

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CBSE Class-10 Revision Notes and Key Points

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