Factorisation Of Polynomials
Factorisation of sum and difference of cubes
\( a^3 + b^3 = (a + b)(a^2 - ab + b^2 ) \)
\( a^3 - b^3 = (a - b)(a^2 + ab + b^2 ) \)
Ex : \( 64a^3 + 27 = (4a)^3 + 3^3 \)
\( = (4a + 3)[(4a)^2 - 4a.3 + 3^2 ) \)
\( = (4a + 3)(16a^2 - 12a + 9) \)
Ex:\( a^6 - b^6 = (a^3 )^2 - (b^3 )^2 \) (\( x^2 - y^2 = (x + y)(x - y) \))
\( = (a^3 - b^3 )(a^3 + b^3 ) \)
\( = (a^3 - b^3 )(a^3 + b^3 ) \)
\( a^6 - b^6 = (a - b)(a^2 + ab + b^2 ) \)
\( (a + b)(a^2 - ab + b^2 ) \)
Factorisation of \( a^3 + b^3 + c^3 - 3abc \) on Simplifying :
\( (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \)
\( = a^3 + ab^2 + ac^2 - a^2 b - abc - a^2 c \)\( + a^2 b + b^3 + c^2 b - ab^2 - b^2 c - abc \)
\( + ca^2 + cb^2 + c^3 - abc - bc^2 - c^2 a \)
\( = a^3 + b^3 + c^3 - 3abc \)
\(\therefore\)\( a^3 + b^3 + c^3 - 3abc \)=\( (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \)
Factorisation of \( a^3 + b^3 + c^3 = 0 \) if a+b+c=0
\( a + b + c = 0 \)
\( \Rightarrow a + b = - c \)
\( \Rightarrow C.O.B.S \)
\( \Rightarrow (a + b)^3 = ( - c)^3 \)
\( \Rightarrow a^3 + b^3 + 3ab(a + b) = - c^3 \)
\( \Rightarrow a^3 + b^3 + 3ab( - c) = - c^3 \)
\( \Rightarrow a^3 + b^3 - 3abc = - c^3 \)
\( \Rightarrow a^3 + b^3 + c^3 = 3abc \)
Ex:\( 5 + 2 - 7 = 0 \Rightarrow 5^3 + 2^3 + ( - 7)^3 = 3.5.( - 7)^3 = 3.5.( - 7).2 = - 210 \)
\( \sum\limits_{a,b,c} a = a + b + c \)
\( \sum\limits_{a,b,c} {a^2 } = a^2 + b^2 + c^2 \)
\( \sum\limits_{a,b,c} {ab} = ab + bc + ca \)
\( \sum\limits_{a,b,c} {(a + b)} = (a + b)(b + c)(c + a) \)
\( \sum\limits_{a,b,c} {a^4 } = a^4 + b^4 + c^4 \)
\( \prod\limits_{a,b,c} a = abc \)
\( \prod\limits_{a,b,c} {a^2 } = a^2 b^2 c^2 \)
1.General form of Quadratic polynomial \( p(x) = ax^2 + bx + c,(a \ne 0) \)
Ex : \( p(x) = \sqrt 3 x^2 - 2x + 5 \)
2. If \( \alpha ,\beta \) are the zeroes of \( p(x) = ax^2 + bx + c, \) then \( ax^2 + bx + c = a(x - \alpha )(x - \beta ) \)
Ex:\( p(x) = 2x^2 + 9x + 10 = 2\left( {x^2 + \frac{9} {2}x + \frac{{10}} {2}} \right) \)
\( = 2\left( {x^2 - \left( { - \frac{9} {2}} \right)x + \frac{{10}} {2}} \right) \)
\( = 2\left( {x^2 - (\alpha + \beta )x + \alpha \beta } \right) = 2(x - \alpha )(x - \beta ) \)
3. If \( \alpha ,\beta \) are the zeroes of \( p(x) = ax^2 + bx + c, \)
Ex :\( p(x) = 3x^2 + 11x + 10 \)
4. If \( \alpha ,\beta \) are the zeroes of quadratic polynomial then
\( p(x) = K[x^2 - (\alpha + \beta )x + \alpha \beta ],K \in R \)
\( p(x) = K[x^2 - (\alpha + \beta )x + \alpha \beta ],K \in R \)Ex : If 2 & 3are the zeroes of a Quadratic polynomial then \(% MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbiqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI % cacaWG4bGaaiykaiabg2da9iaadUeacaGGBbGaamiEamaaCaaaleqa % baGaaGOmaaaakiabgkHiTiaacIcacaaIYaGaey4kaSIaaG4maiaacM % cacaWG4bGaey4kaSIaaGOmaiaac6cacaaIZaGaaiyxaiaacYcacaWG % lbGaeyicI4SaamOuaaaa!4B40! \[ p(x) = K[x^2 - (2 + 3)x + 2.3],K \in R \)
\( p(x) = K[x^2 - 5x + 6],K \in R \)
5. General form of cubic polynomial \( p(x) = ax^3 + bx^2 + cx + d \)
Ex:\( p(x) = 5x^3 - 6x^2 + 3x + 4 \)
6.If \( \alpha ,\beta ,\gamma \) are the zeroes of a cubic polynomial
\( p(x) = ax^3 + bx^2 + cx + d \)
\( S_1 = \alpha + \beta + \gamma = \frac{{ - coefficientofx}} {{coefficientofx^3 }} = - b/a \)
\(% MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbiqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa % aaleaacaaIYaaabeaakiabg2da9iabeg7aHjabek7aIjabgUcaRiab % ek7aIjabeo7aNjabgUcaRiabeo7aNjabeg7aHjabg2da9maalaaaba % acbaGaa83yaiaa-9gacaWFLbGaa8Nzaiaa-zgacaWFPbGaa83yaiaa % -LgacaWFLbGaa8NBaiaa-rhacaWFGaGaaGPaVlaa-9gacaWFMbGaa8 % hiaiaaykW7caWF4baabaGaam4yaiaad+gacaWGLbGaamOzaiaadAga % caWGPbGaam4yaiaadMgacaWGLbGaamOBaiaadshacaaMc8Uaam4Bai % aadAgacaaMc8UaamiEamaaCaaaleqabaGaaG4maaaaaaGccqGH9aqp % daWccaqaaiaadogaaeaacaWGHbaaaaaa!6A9E! S_2 = \alpha \beta + \beta \gamma + \gamma \alpha = \frac{{coefficient \,of \,x}} {{coefficient\,of\,x^3 }} = c/a \)
\( S_3 = \alpha \beta \gamma = \frac{{ - constant}} {{coefficient\,of\,x^3 }} =-d/a \)
Ex : \( 5x^3 + 6x^2 + 7x + 11 \)\( \Rightarrow a = 5,\,b = 6,\,c = 7,\,d = 11 \)
\(
S_2 = \alpha \beta + \beta \gamma + \gamma \alpha = c/a=7/5\)
\(
S_1 = \alpha + \beta + \gamma =-b/a=-6/5\)
\(
S_3 = \alpha \beta \gamma =-d/a=-11/5\)
7. If are the zeroes of \(
p(x) = ax^3 + bx^2 + cx + d
\)
\(
p(x) = ax^3 + bx^2 + cx + d = a(x - \alpha )(x - \beta )(x - \gamma )
\)
\(
= a[x^3 - (\alpha + \beta + \gamma )x^2 + (\alpha \beta + \beta \gamma + \gamma \alpha )x - \alpha \beta \gamma ]
\)
\(
= a[x^3 - S_1 x^2 + S_2 x - S_3 ]
\)
S1 = Sum taken one at a time
S2 = Sum taken 2 at a time
S3 = Sum taken 3 at a time