An algebraic expression is the combination of constants and variable connected by the four basic operations (+, –, ×, ÷).
For example : 2x , x2y , xy/3, 3 etc.
Types of Algebraic expression :
• Polynomial in one variable : An algebraic expression is of the form
are constants and n is non – negative integer.
• Degree of polynomial : largest exponent of the variable is called the degree of the polynomial.
Types of polynomial based on degree
• If deg = 0, then polynomial is called non – zero constant polynomial .
• If deg = 1, then polynomial is called linear polynomial.
• If deg = 2, then polynomial is called quadratic polynomial.
• If deg = 3, then polynomial is called cubic polynomial .
• 0 is called zero polynomial and its deg is not defined .
• Value of a polynomial : Let P(x) be a polynomial, then value of P(x) at x = a is given by P(a) . i.e. by putting x = a.
• Zero of a polynomial : (i) Let P(x) be a polynomial, then x = a is said to be zero of P(x) if P(a) = 0.
(ii) To find zero of P(x), put P(x) = 0 and get the value of x .
• Factor theorem : Let P(x) be a polynomial of deg ≥ 1 and q be any real number then ,
(i) If P(a) = 0, then (x-a) is factor of P(x)
(ii) If (x-a) is factor of P(x), then P(a) = 0.
• Remainder theorem : Let P(x) be a polynomial of deg ≥ 1 and a be any real number. If we divide P(x) by (x-a) then a remainder is given by remainder = P(a)
• Algebraic Identities :
(i) (a+b)2 = a2 + 2ab + b2
(ii) (a-b)2 = a2 – 2ab + b2
(iii) (a2 – b2) = (a – b) (a+b)
(iv) (x+a) (x+b) = x2 + (a+b)x + ab
(v) (a+b+c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
(vi) (a+b)3 = a3 + b3 + 3ab (a+b)
(vii) (a-b)3 = a3 – b3 – 3ab(a-b)
(viii) a3 + b3 = (a+b) (a2 – ab + b2)
(ix) a3 – b3 = (a-b) (a2 + ab + b2)
(x) a3 + b3 + c3 – 3abc = (a+b+c)(a2+b2+c2-ab – bc – ca)