Polynomials
- 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) = an xn + an – 1 xn – 1 + an – 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.
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