Brief Notes Polynomials
1.A polynomial inward i variable is an algebraic aspect of the type
p(x)=anxn+an-1xn-1+…….+a1x+ao , where ao,a1,a2 ……. are constants too n is a positive integer.
Examples: 2x3+3x2-4x+7, 3x4-4x+2, 4x-3 etc
NOTE:
· 2Öx+3 is non a polynomial because the exponent of x inward the showtime term is a fraction.
· x+1/x is non a polynomial because the exponent of x inward the instant term is a negative integer.
2.If an≠0 too then the highest ability of exponent is called score of the polynomial.
Examples:
· The score of 3x3-4x+7 is 3
· The score of 4x2+Ö3x+2 is 2
3. Naming a polynomial according to degree
· Polynomials having score 1 are called linear polynomials, e.g. 2x, 4y-3, 3t+2 etc.
· Polynomials having score two are called quadratic polynomials, e.g.2x2-3x+7,4x2 etc.
· Polynomials having score three 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 two price are called binomials.eg 3t-2,2x2+7,4x3-3 etc.
· Polynomials having three price are called trinomials.eg 2x2-3x+2,3t3-2x+7 etc.
5. A constant polynomial has score 0.
6. The score of cipher polynomial is non defined.
7. To discovery the value of a polynomial p(x) at x=a, lay x=a inward p(x).
8. To discovery the zeroes of a polynomial p(x), lay p(x)=0 too solve.
9. Remainder Theorem
Let P(x) is a polynomial whose score is greater than equal to 1 too ‘a’ endure any real no. When P(x) is divided past times (x-a) too then balance is P(a).
10. Factor theorem- If P(x) is a polynomial whose score is greater than equal to 1 too a endure whatever existent reveal too then (x-a) is a part 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 too then x3+y3+z3=3xyz
9. x3 + y3 = (x+y)(x2-xy+y2)
10. x3 - y3 = (x-y)(x2+xy+y2)
