A polynomial p(x) in one variable x is an algebraic expression of the form, p(x) = anxn + an – 1 xn – 1 + … + a2x2 + a1x + a0, where a0, a1, a2, …, an are constants and an ≠ 0. a0, a1, a2, . . ., an are respectively the coefficients of x0, x, x2, . . ., xn. Each of anxn, an – 1 xn – 1, … a0 is called a term.
An algebraic expression having only one term is called a monomial.
An algebraic expression having two terms is called a binomial.
An algebraic expression having three terms is called a trinomial.
Highest power of the variable in a polynomial is called the degree of the polynomial.
A polynomial whose degree is one is called a linear polynomial.
A polynomial whose degree is two is called a quadratic polynomial.
A polynomial whose degree is three is called a cubic polynomial.
A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0.
Remainder theorem: Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a).
Factor theorem: If p (x) be any polynomial of degree greater than or equal to one and α be any real number, then
(i) (x – α) is a factor of p (x), if p (α) = 0
(ii) p(α) = 0, if (x – α) is a factor of p(x).
Representation of the polynomials as the product of their factors is called factorisation.
Methods of factorisation are:
a) Using factor method
b) Splitting the middle terms
c) Using algebraic identities
Following are the 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 + y3 + 3xy (x + y)
• (x - y)3 = x3 - y3 - 3xy (x - y)
• x3 + y3 + z3 - 3xyz = (x + y + z)(x2 + y2 + z2 – xy –yz – zx)