An algebraic expression is the combination of constants and variable connected by the four basic operations (+, –, ×, ÷).
For example : 2x , x^{2}y , 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} = a^{2 }+ 2ab + b^{2}
(ii) (a-b)^{2} = a^{2} – 2ab + b^{2}
(iii) (a^{2} – b^{2}) = (a – b) (a+b)
(iv) (x+a) (x+b) = x^{2} + (a+b)x + ab
(v) (a+b+c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca
(vi) (a+b)^{3} = a^{3} + b^{3} + 3ab (a+b)
(vii) (a-b)^{3 }= a^{3} – b^{3} – 3ab(a-b)
(viii) a^{3} + b^{3} = (a+b) (a^{2} – ab + b^{2})
(ix) a^{3} – b^{3 }= (a-b) (a^{2} + ab + b^{2})
(x) a^{3 }+ b^{3 }+ c^{3} – 3abc = (a+b+c)(a^{2}+b^{2}+c^{2}-ab – bc – ca)