Math, asked by vibhakhiraiya2007, 1 month ago

factorise y^3-6y^2-19y+84​

Answers

Answered by mathdude500
5

\large\underline{\sf{Solution-}}

Given expression is

\rm :\longmapsto\: {y}^{3} -  {6y}^{2} - 19y + 84

Let assume that

\rm :\longmapsto\: f(y) = {y}^{3} -  {6y}^{2} - 19y + 84

Now, we have to find the zero of f(y) by hit and trial method.

Let y = 1

\rm :\longmapsto\: f(1) = {1}^{3} -  {6(1)}^{2} - 19(1) + 84

\rm :\longmapsto\: f(1) = 1 - 6 - 19 + 84 = 60 \ne \: 0

Let y = 2

\rm :\longmapsto\: f(2) = {2}^{3} -  {6(2)}^{2} - 19(2) + 84

\rm :\longmapsto\: f(2) = 8 - 24 - 38 + 84 = 30 \ne \: 0

Let y = 3

\rm :\longmapsto\: f(3) = {3}^{3} -  {6(3)}^{2} - 19(3) + 84

\rm :\longmapsto\: f(3) =27 -  54 - 57 + 84 = 111 - 111 = 0

\bf\implies \:3 \: is \: the \: zero \: of \: f(y)

\bf\implies \:y - 3 \: is \: the \: factor \: of \: f(y)

So, using long division, we have

\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\:\: {y}^{2} - 3y - 28\:\:}}}\\ {\underline{\sf{y - 3}}}& {\sf{\: {y}^{3}  -  {6y}^{2} - 19y + 84 \:\:}} \\{\sf{}}& \underline{\sf{- {y}^{3} + 3 {y}^{2}   \:   \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:   \:\:}} \\ {{\sf{}}}& {\sf{\:   \:  \:  \:  \:  \:  \:  \:  \:  \:  \: \: -  3{y}^{2} -19y + 84 \:   \:\:}} \\{\sf{}}& \underline{\sf{\:\: \:  \:  \:    \:  \:  \:  3{y}^{2}  - 9y  \:  \:  \:  \:  \:  \: \:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \:  \:  \:  \:  \:   \:  \:  \:  \:  \: \:  - 28y + 84  \:\:}} \\{\sf{}}& \underline{\sf{\: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  28y + 84\:\:}} \\ {\underline{\sf{}}}& {\sf{\:\: \:  \: \:  \:  \:  \:  \:  \:  \:  \: 0\:\:}}  \end{array}\end{gathered}\end{gathered}\end{gathered}

We know,

Dividend = Divisor × Quotient + Remainder

Therefore,

\rm :\longmapsto\: {y}^{3} -  {6y}^{2} - 19y + 84

\rm \:  =  \:(y - 3)( {y}^{2} - 3y - 28)

\rm \:  =  \:(y - 3)( {y}^{2} - 7y + 4y - 28)

\rm \:  =  \:(y - 3)\bigg(y(y - 7) + 4(y - 7)\bigg)

\rm \:  =  \:(y - 3)(y - 7)(y + 4)

Hence,

 \boxed{ \bf{ \: \: {y}^{3} -  {6y}^{2} - 19y + 84 = (y - 3)(y  + 4)(y - 7)}}

Additional Information :-

 \boxed{ \bf{ \:  {(x + y)}^{2} =  {x}^{2} + 2xy +  {y}^{2}}}

 \boxed{ \bf{ \:  {(x  -  y)}^{2} =  {x}^{2}  -  2xy +  {y}^{2}}}

 \boxed{ \bf{ \:  {(x + y)}^{3} =  {x}^{3} + 3xy(x + y) +  {y}^{3}}}

 \boxed{ \bf{ \:  {(x  -  y)}^{3} =  {x}^{3}  -  3xy(x  -  y)  -   {y}^{3}}}

 \boxed{ \bf{ \:  {x}^{3} +  {y}^{3} = (x + y)( {x}^{2} - xy +  {y}^{2})}}

 \boxed{ \bf{ \:  {x}^{3}  -   {y}^{3} = (x  -  y)( {x}^{2}  + xy +  {y}^{2})}}

 \boxed{ \bf{ \:  {x}^{4} -  {y}^{4} = (x - y)(x + y)( {x}^{2} +  {y}^{2})}}

 \boxed{ \bf{ \:  {x}^{2} -  {y}^{2} = (x - y)(x + y)}}

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