Polynomial: A polynomial in one variable x is an algebraic expression of the form
p(x) = a0xn + a1xn-1 + a2xn-2 + … an
Types of Polynomial:
(i) Constant polynomial: A polynomial containing one term only, consisting of a constant is called a constant polynomial. Generally, each real number is a constant polynomial.
(ii) Zero polynomial: A polynomial consisting of one term, namely zero only, is called a zero polynomial.
(iii) Monomial: Polynomials having only one term are called monomials
(iv) Binomial: Polynomials having only two terms are called binomials
(v) Trinomial: Polynomials having only three terms are called trinomials
The degree of a Polynomial: Highest power of the variable in a polynomial is the degree of the polynomial.
(a) In one variable: In case of a polynomial in one variable, the highest power of the variable is called the degree of the polynomial.
(b) In two or more variables: In case of a polynomial in more than one variable, the sum of the powers of the variables in each term is taken up and the highest sum so obtained is called the degree of the polynomial.
Linear Polynomial: A polynomial of degree one is called a linear polynomial. e.g.,
x + 5 is a linear polynomial in x, y and z.
Quadratic Polynomial: A polynomial of degree two is called a quadratic polynomial. e.g.,
2x2 + x+ 1
Cubic Polynomial: A polynomial of degree three is called a cubic polynomial. e.g.,
2y3 + 3
Value of a Polynomial: Value of a polynomial p(x) at x = a is p(a).
e.g., If p(x) = x2 + 2x + 6 then, at x = 2, p(2) = 22 + 2 × 2 + 6 = 14
Zeroes of a Polynomial: Zeroes of a polynomial p(x) is a number a such that p(a) = 0.
- Zero may be a zero of a polynomial.
- Every linear polynomial has one and only one zero.
- Zero of a polynomial is also called the root of the polynomial.
- A non-zero constant polynomial has no zero.
- Every real number is a zero of the zero polynomial.
- A polynomial can have more than one zero.
The maximum number of zeroes of a polynomial is equal to its degree.
Remainder Theorem: Let p (x) be any polynomial of degree n greater than or equal to one (n ≥ 1) and let a be any real number. If p (x) is divided by the linear polynomial x – a, then the remainder is p (a).
Dividend = (Divisor × Quotient) + Remainder
Factor Theorem: Let q(x) be a polynomial of degree n ≥ 1 and a be any real number, then
(i) (x – a) is a factor of q (x), if q(a) = 0 and
(ii) q(a) = 0, if x – a is a factor of q (x).
Algebraic Identities: An algebraic identity is an algebraic equation that is true for all values of the variable occurring in it.
Some algebraic identities are given below
- (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)
If x + y + z = 0, then x3 + y3 + z3 = 3xyz