POLYNOMIAL - ESSENTIAL POINTS
- A polynomial p(x) in one variable x is an algebraic expression in x of the form p(x) = anxn + an–1xn – 1 + . . . + a2x2 + a1x + a0,
- a0, a1, a2, . . ., an are respectively the coefficients of x0, x, x2, . . ., xn, and n is called the degree of the polynomial.
- Each of anxn, an–1 xn–1, ..., a0, with an ≠ 0, is called a term of the polynomial p(x).
- A polynomial of one term is called a monomial, two terms is called a binomial and three terms is called a trinomial.
- A polynomial of degree one is called a linear polynomial, degree two is called a quadratic polynomial and degree three is called a cubic polynomial.
- A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0. In this case, a is also called a root of the equation p(x) = 0.
- Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial has no zero, and every real number is a zero of the zero polynomial.
- Remainder Theorem : If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial x – a, then the remainder is p(a).
- Factor Theorem : x – a is a factor of the polynomial p(x), if p(a) = 0. Also, if x – a is a factor of p(x), then p(a) = 0.
where a0, a1, a2, . . ., an are constants and an ≠ 0.
Algebraic identities –
(x + y)2 = x2 + 2xy + y2
(x - y)2 = x2 - 2xy + y2
x2 – y2 = (x + y) (x – y)
(x + a) (x + b) = x2 + (a + b) x + ab
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
(x + y)3 = x3 + 3x2y + 3xy2 + y3 = x3 + y3 + 3xy (x + y)
(x – y)3 = x3 – 3x2y + 3xy2 – y3 = x3 – y3 – 3xy (x – y)
x3 + y3 = (x + y) (x2 – xy + y2)
x3 – y3 = (x – y) (x2 + xy + y2)
x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)
