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22 July, 2015

Brief notes- Polynomials

Brief Notes Polynomials
1.A polynomial in one variable is an algebraic expression of the type
p(x)=anxn+an-1xn-1+…….+a1x+ao    , where ao,a1,a……. are constants and n is a positive integer.
Examples:  2x3+3x2-4x+7, 3x4-4x+2, 4x-3 etc
NOTE:
·        2Öx+3 is not a polynomial because the exponent of x in the first term is a fraction.
·        x+1/x is not a polynomial because the exponent of x in the second term is a negative integer.
2.If an≠0 then the highest power of exponent is called degree of the polynomial.
Examples:
·        The degree of 3x3-4x+7 is 3
·        The degree of 4x2+Ö3x+2 is 2
3. Naming a polynomial according to degree
·        Polynomials having degree 1 are called linear polynomials, e.g. 2x, 4y-3, 3t+2 etc.
·        Polynomials having degree 2 are called quadratic polynomials, e.g.2x2-3x+7,4x2 etc.
·        Polynomials having degree 3 are called cubic polynomials, e.g. 3x3-2x+7, 3x3-2 etc.
4. Naming a polynomial according to no. of terms
·        Polynomials having 1 term are called monomials.eg 2x,3t,7 etc.
·        Polynomials having 2 terms are called binomials.eg 3t-2,2x2+7,4x3-3 etc.
·        Polynomials having 3 terms are called trinomials.eg 2x2-3x+2,3t3-2x+7 etc.
5.  A constant polynomial has degree 0.
6. The degree of zero polynomial is not defined.
7. To find the value of a polynomial p(x) at x=a, put x=a in p(x).
8. To find the zeroes of a polynomial p(x), put p(x)=0 and solve.
9. Remainder Theorem
Let P(x) is a polynomial whose degree is greater than equal to 1 and ‘a’ be any real no. When P(x) is divided by (x-a) then remainder is P(a).
10. Factor theorem- If P(x) is a polynomial whose degree is greater than equal to 1 and a be any real number then (x-a) is a factor of P(x) when p(a)=0
List of Formulae
1.     (x+y)2=x2+2xy+y2
2.     (x-y)2=x2-2xy+y2
3.     X2-y2=(x-y)(x+y)
4.     (x+a)(x+b)=x2+(a+b)x+ab
5.     (x+y+z)2=x2+y2+z2+2xy+2yz+2zx
6.     (x+y)3=x3+3x2y+3xy2+y3
OR  x3+3xy(x+y)+y3
7.     (x-y)3=x3-3x2y+3xy2-y3
OR  x3-3xy(x-y)-y3
8.     x3+y3+z3-3xyz=(x+y+z)(x2+y2+z2-xy-yz-zx)
When (x+y+z)=0 then x3+y3+z3=3xyz
9.     x3 + y3 = (x+y)(x2-xy+y2)
10.                        x3 - y3 = (x-y)(x2+xy+y2)