Polynomials

02 Polynomials

POLYNOMIALS

 

Polynomial: Polynomial with variable x, is denoted by p(x) is expression of the type

a n x n  + a x n-1 n-1 + a x n-2 n-2  +... + a 1 x + a 0 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabggadaWgaa WcbaGaaeOBaaqabaGccaWG4bWaaWbaaSqabeaacaqGUbaaaOGaaeii aiaabUcacaqGGaGaaeyyamaaBeaaleaacaqGUbGaaeylaiaabgdaae qaaOGaamiEamaaCaaaleqabaGaaeOBaiaab2cacaqGXaaaaOGaae4k aiaabccacaqGHbWaaSraaSqaaiaab6gacaqGTaGaaeOmaaqabaGcca WG4bWaaWbaaSqabeaacaqGUbGaaeylaiaabkdaaaGccaqGGaGaae4k aiaab6cacaqGUaGaaeOlaiaabUcacaqGGaGaaeyyamaaBaaaleaaca qGXaaabeaakiaabIhacaqGGaGaae4kaiaabccacaqGHbWaaSbaaSqa aiaabcdaaeqaaOGaaeOlaaaa@582D@

a n ,  a n-1 , a n-2 ,  ... ,  a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabggadaWgaa WcbaGaaeOBaaqabaGccaqGSaGaaeiiaiaabccacaqGHbWaaSbaaSqa aiaab6gacaqGTaGaaeymaaqabaGccaqGSaGaaeiiaiaabggadaWgaa WcbaGaaeOBaiaab2cacaqGYaaabeaakiaabYcacaqGGaGaaeiiaiaa b6cacaqGUaGaaeOlaiaabYcacaqGGaGaaeiiaiaabggadaWgaaWcba Gaaeimaaqabaaaaa@4AC5@ are the coeeficients, and x is variable. a 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabggadaWgaa WcbaGaaeimaaqabaaaaa@3897@ is called constant term.

The polynomial p( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jc9qqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiCa8aadaqadaqaa8qacaqG4baapaGaayjkaiaawMcaaaaa@3A77@ has n + 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jc9qqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaab6gacaqGGa Gaae4kaiaabccacaqGXaaaaa@3A4B@ terms.

e.g. x 5  - 5x 4  + 10x 3 +7x 2 -8x + 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaahaa WcbeqaaiaabwdaaaGccaqGGaGaaeylaiaabccacaqG1aGaaeiEamaa CaaaleqabaGaaGinaaaakiaabccacaqGRaGaaeiiaiaabgdacaqGWa GaaeiEamaaCaaaleqabaGaaG4maaaakiaabUcacaqG3aGaaeiEamaa CaaaleqabaGaaGOmaaaakiaab2cacaqG4aGaaeiEaiaabccacaqGRa Gaaeiiaiaabsdaaaa@4B0F@

 

 

Monomials: A polynomial with one term is called monomial.

e.g 2x,  3x 3 ,  -5y 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabkdacaqG4b GaaeilaiaabccacaqGGaGaae4maiaabIhadaahaaWcbeqaaiaaboda aaGccaqGSaGaaeiiaiaabccacaqGTaGaaeynaiaabMhadaahaaWcbe qaaiaabAdaaaaaaa@4256@

 

 

Binomial: A polynomial with two terms is called binomial.

e.g. 3x + 3x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabodacaqG4b GaaeiiaiaabUcacaqGGaGaae4maiaabIhadaahaaWcbeqaaiaabkda aaaaaa@3D0C@

 

 

Trinomial: A polynomial with 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jc9qqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4maaaa@3788@ terms is called trinomial.

e.g. 3x + 3x 2 + 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabodacaqG4b GaaeiiaiaabUcacaqGGaGaae4maiaabIhadaahaaWcbeqaaiaabkda aaGccaqGRaGaaeiiaiaabAdaaaa@3F20@

 

 

Constant polynomial: Aconstant polynomial isa polynomial which is free of variable,and just contains a constant term

e.g. p( x ) = 5, p( y ) = -7, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jc9qqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiCa8aadaqadaqaa8qacaqG4baapaGaayjkaiaawMcaa8qacaqG GaGaaeypaiaabccacaqG1aGaaeilaiaabccacaqGWbWdamaabmaaba WdbiaabMhaa8aacaGLOaGaayzkaaWdbiaabccacaqG9aGaaeiiaiaa b2cacaqG3aGaaeilaaaa@466C@

 

Degree of polynomial: It is the highest power of variable in the expression of polynomial p(x).

The Polynomial p( x )  = x 5  - 3x 2  + 2x + 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabchadaqada qaaiaabIhaaiaawIcacaGLPaaacaqGGaGaaeypaiaabccacaqG4bWa aWbaaSqabeaacaqG1aaaaOGaaeiiaiaab2cacaqGGaGaae4maiaabI hadaahaaWcbeqaaiaabkdaaaGccaqGGaGaae4kaiaabccacaqGYaGa aeiEaiaabccacaqGRaGaaeiiaiaabsdaaaa@491D@ has degree 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jc9qqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeynaaaa@378A@ .

 

 

Cubic polynomial: A polynomial with degree 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jc9qqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4maaaa@3788@ is called cubic polynomial.

e.g. x 3  - 3x 2  + 2x + 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabIhadaahaa WcbeqaaiaabodaaaGccaqGGaGaaeylaiaabccacaqGZaGaaeiEamaa CaaaleqabaGaaeOmaaaakiaabccacaqGRaGaaeiiaiaabkdacaqG4b GaaeiiaiaabUcacaqGGaGaaeinaaaa@439E@

 

 

Quadratic Polynomial: A polynomial with degree 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jc9qqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOmaaaa@3787@ is called Quadratic polynomial.

e.g. 3x 2  + 2x + 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabodacaqG4b WaaWbaaSqabeaacaqGYaaaaOGaaeiiaiaabUcacaqGGaGaaeOmaiaa bIhacaqGGaGaae4kaiaabccacaqG0aaaaa@3FC0@

 

 

Linearpolynomial : A polynomial with degree 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jc9qqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeymaaaa@3786@ is called Linear polynomial.

e.g. 2x+4

Degree of constant polynomial is zero.

 

 

Value of polynomial:

Value obtained after replacing x by a real number k in the expression of p( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jc9qqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiCa8aadaqadaqaa8qacaqG4baapaGaayjkaiaawMcaaaaa@3A77@ is called value of polynomial at x = k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Jc9qqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiEaiaabccacaqG9aGaaeiiaiaabUgaaaa@3AC1@ .

For p(x)= x 2  - 14x + 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabIhadaahaa WcbeqaaiaabkdaaaGccaqGGaGaaeylaiaabccacaqGXaGaaeinaiaa bIhacaqGGaGaae4kaiaabccacaqGZaGaae4maaaa@4077@

Put x=4

We get p(4) = 4 2  - 14 × 4 + 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabsdadaahaa WcbeqaaiaabkdaaaGccaqGGaGaaeylaiaabccacaqGXaGaaeinaiaa bccacaqGxdGaaeiiaiaabsdacaqGGaGaae4kaiaabccacaqGZaGaae 4maaaa@428F@

= 16 – 56 + 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeypaiaabccacaqGXaGaaeOnaiaabccacaqGtaIaaeiiaiaabwda caqG2aGaaeiiaiaabUcacaqGGaGaae4maiaabodaaaa@4091@

p( 4 ) = -7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabchadaqada qaaiaabsdaaiaawIcacaGLPaaacaqGGaGaaeypaiaabccacaqGTaGa ae4naaaa@3D77@ .

We say -7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaab2cacaqG3a aaaa@383E@ is the value of polynomial at x = 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiEaiaabccacaqG9aGaaeiiaiaabsdaaaa@3AAC@ .

 

Zeroes of polynomial:The values of x, at which the polynomial p(x) is equal to 0 is called as zeroes of polynomial.

For e.g. p( x )  = x 2  - 14x + 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabchadaqada qaaiaabIhaaiaawIcacaGLPaaacaqGGaGaaeypaiaabccacaqG4bWa aWbaaSqabeaacaqGYaaaaOGaaeiiaiaab2cacaqGGaGaaeymaiaabs dacaqG4bGaaeiiaiaabUcacaqGGaGaae4maiaabodaaaa@45F4@

We get, p( 11 )  = 11 2  - 14 × 11 + 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabchadaqada qaaiaabgdacaqGXaaacaGLOaGaayzkaaGaaeiiaiaab2dacaqGGaGa aeymaiaabgdadaahaaWcbeqaaiaabkdaaaGccaqGGaGaaeylaiaabc cacaqGXaGaaeinaiaabccacaqGxdGaaeiiaiaabgdacaqGXaGaaeii aiaabUcacaqGGaGaae4maiaabodaaaa@49DB@

= 121 - 154 + 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeypaiaabccacaqGXaGaaeOmaiaabgdacaqGGaGaaeylaiaabcca caqGXaGaaeynaiaabsdacaqGGaGaae4kaiaabccacaqGZaGaae4maa aa@41ED@

p( 11 ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiCa8aadaqadaqaa8qacaqGXaGaaeymaaWdaiaawIcacaGLPaaa peGaaeiiaiaab2dacaqGGaGaaeimaaaa@3DCF@ So 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeymaiaabgdaaaa@385C@ is called zero of the given polynomial.

 

 

Remainder theorem:

When polynomial p( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiCa8aadaqadaqaa8qacaqG4baapaGaayjkaiaawMcaaaaa@3A99@ is divided by polynomial Q( x ) = x - a ( deg Q( x ) < deg p( x ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaeyua8aadaqadaqaa8qacaqG4baapaGaayjkaiaawMcaa8qacaqG GaGaaeypaiaabccacaqG4bGaaeiiaiaab2cacaqGGaGaaeyyaiaabc capaWaaeWaaeaapeGaaeizaiaabwgacaqGNbGaaeiiaiaabgfapaWa aeWaaeaapeGaaeiEaaWdaiaawIcacaGLPaaapeGaaeiiaiaabYdaca qGGaGaaeizaiaabwgacaqGNbGaaeiiaiaabchapaWaaeWaaeaapeGa aeiEaaWdaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@52A8@ , then the remainder will be p(a).

e.g.

p( x )  = 3x 3  - 4x 2  + 4x + 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabchadaqada qaaiaabIhaaiaawIcacaGLPaaacaqGGaGaaeypaiaabccacaqGZaGa aeiEamaaCaaaleqabaGaae4maaaakiaabccacaqGTaGaaeiiaiaabs dacaqG4bWaaWbaaSqabeaacaqGYaaaaOGaaeiiaiaabUcacaqGGaGa aeinaiaabIhacaqGGaGaae4kaiaabccacaqG1aaaaa@49D5@

q( x ) = x - 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyCa8aadaqadaqaa8qacaqG4baapaGaayjkaiaawMcaa8qacaqG GaGaaeypaiaabccacaqG4bGaaeiiaiaab2cacaqGGaGaaeOmaaaa@4056@

If p( x ) ÷ q( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabchadaqada qaaiaabIhaaiaawIcacaGLPaaacaqGGaGaae49aiaabccacaqGXbWa aeWaaeaacaqG4baacaGLOaGaayzkaaaaaa@4083@

Remainder= p( 2 )  = 3 × 2 3  - 4 × 2 2  + 4 × 2 + 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabchadaqada qaaiaabkdaaiaawIcacaGLPaaacaqGGaGaaeypaiaabccacaqGZaGa aeiiaiaabEnacaqGGaGaaeOmamaaCaaaleqabaGaae4maaaakiaabc cacaqGTaGaaeiiaiaabsdacaqGGaGaae41aiaabccacaqGYaWaaWba aSqabeaacaqGYaaaaOGaaeiiaiaabUcacaqGGaGaaeinaiaabccaca qGxdGaaeiiaiaabkdacaqGGaGaae4kaiaabccacaqG1aaaaa@509D@

= 24 – 16 + 8 + 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeypaiaabccacaqGYaGaaeinaiaabccacaqGtaIaaeiiaiaabgda caqG2aGaaeiiaiaabUcacaqGGaGaaeioaiaabccacaqGRaGaaeiiai aabwdaaaa@4287@

= 21 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeypaiaabccacaqGYaGaaeymaaaa@39C0@

 

 

Factor theorem:

For polynomial p( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiCa8aadaqadaqaa8qacaqG4baapaGaayjkaiaawMcaaaaa@3A99@ , if p( a ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiCa8aadaqadaqaa8qacaqGHbaapaGaayjkaiaawMcaa8qacaqG GaGaaeypaiaabccacaqGWaaaaa@3D4B@ , then ( x - a ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaaeaa aaaaaaa8qacaqG4bGaaeiiaiaab2cacaqGGaGaaeyyaaWdaiaawIca caGLPaaaaaa@3C61@ is factor of p( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiCa8aadaqadaqaa8qacaqG4baapaGaayjkaiaawMcaaaaa@3A99@

e.g.For p( x )  = x 2  - 14x + 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiCa8aadaqadaqaa8qacaqG4baapaGaayjkaiaawMcaaiaabcca caqG9aGaaeiiaiaabIhadaahaaWcbeqaaiaabkdaaaGccaqGGaGaae ylaiaabccacaqGXaGaaeinaiaabIhacaqGGaGaae4kaiaabccacaqG ZaGaae4maaaa@4642@

p( 3 ) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiCa8aadaqadaqaa8qacaqGZaaapaGaayjkaiaawMcaa8qacaqG GaGaaeypaiaabccacaqGWaaaaa@3D1D@

so(x-3) is factor of x 2  - 14x + 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabIhadaahaa WcbeqaaiaabkdaaaGccaqGGaGaaeylaiaabccacaqGXaGaaeinaiaa bIhacaqGGaGaae4kaiaabccacaqGZaGaae4maaaa@4077@ .

 

Algebraic identities:

1) ( a + b ) 2  = a 2  + 2ab + b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaae yyaiaabccacaqGRaGaaeiiaiaabkgaaiaawIcacaGLPaaadaahaaWc beqaaiaabkdaaaGccaqGGaGaaeypaiaabccacaqGHbWaaWbaaSqabe aacaqGYaaaaOGaaeiiaiaabUcacaqGGaGaaeOmaiaabggacaqGIbGa aeiiaiaabUcacaqGGaGaaeOyamaaCaaaleqabaGaaeOmaaaaaaa@4909@

2) ( a - b ) 2  = a 2  - 2ab + b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaae yyaiaabccacaqGTaGaaeiiaiaabkgaaiaawIcacaGLPaaadaahaaWc beqaaiaabkdaaaGccaqGGaGaaeypaiaabccacaqGHbWaaWbaaSqabe aacaqGYaaaaOGaaeiiaiaab2cacaqGGaGaaeOmaiaabggacaqGIbGa aeiiaiaabUcacaqGGaGaaeOyamaaCaaaleqabaGaaeOmaaaaaaa@490D@

3) ( a + b )( a - b )  = a 2  - b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaae yyaiaabccacaqGRaGaaeiiaiaabkgaaiaawIcacaGLPaaadaqadaqa aiaabggacaqGGaGaaeylaiaabccacaqGIbaacaGLOaGaayzkaaGaae iiaiaab2dacaqGGaGaaeyyamaaCaaaleqabaGaaeOmaaaakiaabcca caqGTaGaaeiiaiaabkgadaahaaWcbeqaaiaabkdaaaaaaa@48F5@

4) ( x + a )( x + b )  = x 2  + ( a + b )x + ab  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaae iEaiaabccacaqGRaGaaeiiaiaabggaaiaawIcacaGLPaaadaqadaqa aiaabIhacaqGGaGaae4kaiaabccacaqGIbaacaGLOaGaayzkaaGaae iiaiaab2dacaqGGaGaaeiEamaaCaaaleqabaGaaeOmaaaakiaabcca caqGRaGaaeiiamaabmaabaGaaeyyaiaabccacaqGRaGaaeiiaiaabk gaaiaawIcacaGLPaaacaqG4bGaaeiiaiaabUcacaqGGaGaaeyyaiaa bkgacaqGGaaaaa@520F@

5) ( a + b + c ) 2  = a 2  + b 2  + c 2  + 2ab + 2bc + 2ac MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaae yyaiaabccacaqGRaGaaeiiaiaabkgacaqGGaGaae4kaiaabccacaqG JbaacaGLOaGaayzkaaWaaWbaaSqabeaacaqGYaaaaOGaaeiiaiaab2 dacaqGGaGaaeyyamaaCaaaleqabaGaaeOmaaaakiaabccacaqGRaGa aeiiaiaabkgadaahaaWcbeqaaiaabkdaaaGccaqGGaGaae4kaiaabc cacaqGJbWaaWbaaSqabeaacaqGYaaaaOGaaeiiaiaabUcacaqGGaGa aeOmaiaabggacaqGIbGaaeiiaiaabUcacaqGGaGaaeOmaiaabkgaca qGJbGaaeiiaiaabUcacaqGGaGaaeOmaiaabggacaqGJbaaaa@589A@

5) ( a + b ) 3  = a 3  + 3a 2 b + 3ab 2  + b 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaae yyaiaabccacaqGRaGaaeiiaiaabkgaaiaawIcacaGLPaaadaahaaWc beqaaiaabodaaaGccaqGGaGaaeypaiaabccacaqGHbWaaWbaaSqabe aacaqGZaaaaOGaaeiiaiaabUcacaqGGaGaae4maiaabggadaahaaWc beqaaiaabkdaaaGccaqGIbGaaeiiaiaabUcacaqGGaGaae4maiaabg gacaqGIbWaaWbaaSqabeaacaqGYaaaaOGaaeiiaiaabUcacaqGGaGa aeOyamaaCaaaleqabaGaae4maaaaaaa@4F58@

Or = a 3  + 3ab( a + b )  + b 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabggadaahaa WcbeqaaiaabodaaaGccaqGGaGaae4kaiaabccacaqGZaGaaeyyaiaa bkgadaqadaqaaiaabggacaqGGaGaae4kaiaabccacaqGIbaacaGLOa GaayzkaaGaaeiiaiaabUcacaqGGaGaaeOyamaaCaaaleqabaGaae4m aaaaaaa@461A@

6) ( a - b ) 3  = a 3  - 3a 2 b + 3ab 2  - b 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaae yyaiaabccacaqGTaGaaeiiaiaabkgaaiaawIcacaGLPaaadaahaaWc beqaaiaabodaaaGccaqGGaGaaeypaiaabccacaqGHbWaaWbaaSqabe aacaqGZaaaaOGaaeiiaiaab2cacaqGGaGaae4maiaabggadaahaaWc beqaaiaabkdaaaGccaqGIbGaaeiiaiaabUcacaqGGaGaae4maiaabg gacaqGIbWaaWbaaSqabeaacaqGYaaaaOGaaeiiaiaab2cacaqGGaGa aeOyamaaCaaaleqabaGaae4maaaaaaa@4F5E@

Or = a 3  - 3ab( a - b )  - b 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabggadaahaa WcbeqaaiaabodaaaGccaqGGaGaaeylaiaabccacaqGZaGaaeyyaiaa bkgadaqadaqaaiaabggacaqGGaGaaeylaiaabccacaqGIbaacaGLOa GaayzkaaGaaeiiaiaab2cacaqGGaGaaeOyamaaCaaaleqabaGaae4m aaaaaaa@4620@

7) a 3  + b 3  + c 3  - 3abc = ( a + b + c )( a 2  + b 2  + c 2  - ab - bc - ac ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabggadaahaa WcbeqaaiaabodaaaGccaqGGaGaae4kaiaabccacaqGIbWaaWbaaSqa beaacaqGZaaaaOGaaeiiaiaabUcacaqGGaGaae4yamaaCaaaleqaba Gaae4maaaakiaabccacaqGTaGaaeiiaiaabodacaqGHbGaaeOyaiaa bogacaqGGaGaaeypaiaabccadaqadaqaaiaabggacaqGGaGaae4kai aabccacaqGIbGaaeiiaiaabUcacaqGGaGaae4yaaGaayjkaiaawMca amaabmaabaGaaeyyamaaCaaaleqabaGaaeOmaaaakiaabccacaqGRa GaaeiiaiaabkgadaahaaWcbeqaaiaabkdaaaGccaqGGaGaae4kaiaa bccacaqGJbWaaWbaaSqabeaacaqGYaaaaOGaaeiiaiaab2cacaqGGa GaaeyyaiaabkgacaqGGaGaaeylaiaabccacaqGIbGaae4yaiaabcca caqGTaGaaeiiaiaabggacaqGJbaacaGLOaGaayzkaaaaaa@65D7@

8) a 3  + b 3  = ( a + b )( a 2  - ab + b 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabggadaahaa WcbeqaaiaabodaaaGccaqGGaGaae4kaiaabccacaqGIbWaaWbaaSqa beaacaqGZaaaaOGaaeiiaiaab2dacaqGGaWaaeWaaeaacaqGHbGaae iiaiaabUcacaqGGaGaaeOyaaGaayjkaiaawMcaamaabmaabaGaaeyy amaaCaaaleqabaGaaeOmaaaakiaabccacaqGTaGaaeiiaiaabggaca qGIbGaaeiiaiaabUcacaqGGaGaaeOyamaaCaaaleqabaGaaeOmaaaa aOGaayjkaiaawMcaaaaa@4E94@

9) a 3  - b 3  = ( a - b )( a 2  + ab + b 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqpG0df9frFj0=yqpe ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabggadaahaa WcbeqaaiaabodaaaGccaqGGaGaaeylaiaabccacaqGIbWaaWbaaSqa beaacaqGZaaaaOGaaeiiaiaab2dacaqGGaWaaeWaaeaacaqGHbGaae iiaiaab2cacaqGGaGaaeOyaaGaayjkaiaawMcaamaabmaabaGaaeyy amaaCaaaleqabaGaaeOmaaaakiaabccacaqGRaGaaeiiaiaabggaca qGIbGaaeiiaiaabUcacaqGGaGaaeOyamaaCaaaleqabaGaaeOmaaaa aOGaayjkaiaawMcaaaaa@4E96@

 

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