Polynomials Class 10 Notes Maths: Chapter 2

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Polynomialsa

An algebraic expression, in which variables do not occur in the denominator, powers of variables are whole numbers and numerical coefficients of various terms are real numbers, iscalled a polynomial.
General representation of a polynomial: f(x) = a_n xⁿ + an – 1 x^{n – 1} + a_{n – 2} xn– 2+…….+ a₁x + a₀ where, n = positive integer and the constants a₀, a₁, a₂…are called as coefficients. Here f(x) is a polynomial invariable x.
We can do the following operations while solving polynomials
(i) Addition of polynomials
(ii) Subtraction of polynomials
(iii) Multiplication of polynomials
(iv) Division of Polynomials
Polynomials can be classified on the basis of number of terms as :
(i) Monomial : One term, for example : 2x, x², 2x³
(ii) Binomial : Two terms, for example : 6a – 5, x²+1
(iii) Trinomial : Three terms, for example : m²– m – 1, x+y+z4
The highest value of exponents is called degree of polynomial. For example : x² – 3 is a second degree polynomial. a + 11 is a first degree polynomial. x⁴ – 16 is a fourth degree polynomial. Also, a constant term (such as 7,22, – 9) is a zero degree polynomial.
A polynomial of degree 0 is called a constant polynomial
If α and β are the zeroes of p(x) = ax² + bx + c and a ≠ 0, then
(i) α + β = – b/a, and
(ii) α β = c/a
A quadratic polynomial whose zeroes are α and β is given byp(x) or f(x) = x² – (α + β)x + αβ.
If α, β and γ are the zeroes of polynomial p(x)=ax³ + bx² + cx + d and a ≠ 0, then
(i) α + β + γ = –b/a
(ii) αβ + βγ + γα =ca
(iii) αβγ = –d/a
A cubic polynomial whose zeroes are α, β and γ is given byp(x) or f(x) = x³ – (α + β + γ)x² + (αβ + βγ + γα)x – αβγ
A real number ‘a’ is a zero of a polynomialf(x) if f(a) = 0
A polynomial ofdegree n can have atmostn real roots.
Geometrically, the zeroes of a polynomial f(x) are the x-coordinates of the points where the graphy= f(x) intersects X-axis.