Polynomials - Revision Notes

 CBSE Class–10 Mathematics

Revision Notes
CHAPTER 02
POLYNOMIALS

1. ---

   Geometrical Meaning of the Zeroes of a Polynomial
2. Zeroes and Coefficients of a Polynomial
3. Division Algorithm for Polynomials

1. Monomials: Algebraic expression with one term is known as Monomial.
2. Binomial: Algebraic expression with two terms is called Binomial.
3. Trinomial: Algebraic expression with three terms is known as Trinomial.
4. Polynomials: All above mentioned algebraic expressions are called Polynomials.
5. Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
6. A quadratic polynomial in x with real coefficient is of the form ax2 + bx + c$\text{a}{\text{x}}^2+\text{ bx }+\text{ c}$,where a, b, c are real numbers with a ≠0$\text{a }\ne 0$.
7. 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.
8. A polynomial can have at most the same number of zeros as the degree of polynomial.
9. For quadratic polynomial ax2+ bx + c$\text{a}{\text{x}}^2+\text{ bx }+\text{ c}$(a≠0)$(a\ne 0)$

Sum of zeroes =−ba$-\frac{b}{a}$

Product of zeroes = ca$\frac{c}{a}$

10. In a cubic polynomial ax3+bx2+cx+d$a{x}^3+b{x}^2+cx+d$, if α,β,γ$\alpha ,\beta ,\gamma$ are the zeroes of the polynomial, then

α+β+γ=−ba$\alpha +\beta +\gamma =\frac{-b}{a}$

αβ+βγ+γα=ca$\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a}$

α.β.γ=da$\alpha .\beta .\gamma =\frac{d}{a}$

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 :

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

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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