Question

If the zeroes of the polynomial x 3 – 3x 2 + x + 1 are a – b, a, a + b, then find the values of a and b.

Answer 1

Answer:

answer for the given problem is given

Answer 2

$\huge{ \underline{\underline{\mathrm{ \red{Question}}}}}$

If the zeroes of the polynomial x 3 – 3x 2 + x + 1 are a – b, a, a + b, then find the values of a and b.

$\huge{ \underline{\underline{\mathrm{ \red{Answer}}}}}$

$\sf \underline{\: The \: zero \: of \: a \: polynomial \: is \: a \: value \: of \: the \: variable \: at \: which \: the \: polynomial \: equals \: 0.}$

$\sf \underline{ \: For \: the \: said \: polynomial, \: we \: have \: these \: equations:}$

$\sf \: a^ 3−3a^2+a+1=0 \\ { \sf</p><p>(a−b)^3−3(a−b)^2+a−b+1=0 }\\ { \sf</p><p>or,a^3−3a^2b+3ab^2−b^3−3a^2+6ab−3b^2+a−b+1=0}</p><p>$

$\sf \: 3.(a+b) ^ 3−3(a+b)^2+a+b+1=0 \\ </p><p>{ \sf</p><p>or,a^3+3a^2b+3ab^2+b^3−3a^2−6ab−3b^2+a+b+1=0}</p><p>$

${ \sf{ \underline{Using \: equations \: 1 \: and \: 2, \: we \: get,}}}$

$\sf \: −3a2b+3ab2−b3−3b2+6ab−b=0 \\ { \sf−3a2b+3ab2−b3−3b2+6ab−b=0 }</p><p> \\ { \sf</p><p>or, 3a2b−3ab2+b3+3b2−6ab+b=0 }$

$\sf{ \underline{Using \: equations \: 1 \: and \: 3, \: we \: get,}}$

$\sf \: 3a ^ 2b+3ab^2+b^3−3b^2−6ab+b=0$

$\sf{ \underline{Subtracting \: last \: two \: equations, \: we \: get,}}$

$\sf \: 6ab ^ 2−6b^2=0 \\ { \sf</p><p></p><p>or, 6b^2(a−1)=0} \\ { \sf</p><p></p><p>or, b=0 or a=1}</p><p>$

$\sf \underline{Now, \: putting \: a=1 \: in \: either \: equation \: 1 \: or \: equation \: 2, \: we \: get,}$

$\sf \: 3b+3b ^ 2+b^3−3b^2−6b+b=0 \\ { \sf</p><p></p><p>or, b^3−2b=0} \\ { \sf</p><p></p><p>or, b(b^2−2)=0} \\ { \sf</p><p></p><p>or, b=0 , or b=√2 , or b = -√2 }</p><p>$

$\sf{ \underline{So, \: we \: have \: got \: a=1 \: and \: b=0,√2 , -√2}}$

$\bold{ \underline{ \underline{Therefore, \: the \: zeroes \: of \: the \: said \: polynomial \: are \: 1,1+√2 \: and \: 1-√2}}}$