Question

what must be added to 9x^3-6x^2+6x-7 to make the sum -6x^3-3x^2-x+1


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Answer 1

let the no. that will be added be x

• $\sf 9x³-6x²+6x-7+x = -6x³-3x²-x+1$

• $\sf x = -6x³-3x²-x+1-(9x³-6x²+6x-7)$

• $\sf x = -6x³-3x²-x+1-9x³+6x²-6x+7$

• $\sf x = -15x³+3x²-7x+8$

Verification:-

• $\sf 9x³-6x²+6x-7+(-15x³+3x²-7x+8)$

• $\sf 9x³-6x²+6x-7-15x³+3x²-7x+8$

• $\sf -6x³-3x²-x+1$

hence, the required polynomial is -6x³-3x²-x+1

Answer 2

$\large\underline\purple{\bold{Solution :- }}$

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$\longrightarrow \red{ \bf \: Let \: y \: will \: be \: added.}$

$\tt {9x}^{3} - {6x}^{2} + 6x - 7 + \: y = { - 6x}^{3} - {3x}^{2} - x + 1$

$\tt \: y = { - 6x}^{3} - {3x}^{2} - x + 1 - ({9x}^{3} - {6x}^{2} + 6x - 7)$

$\tt \: y = { - 6x}^{3} - {3x}^{2} - x + 1 - {9x}^{3} + {6x}^{2} - 6x + 7$

$\boxed{ \red{ \tt \: y = - {15x}^{3} + {3x}^{2} - 7x + 8}}$

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$\large \boxed{ \pink{ \tt \: Verification : - }}$

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Consider LHS

$\tt \: {9x}^{3} - \: {6x}^{2} + \: 6x - 7 \: + \: y$

$\tt \: {9x}^{3} - {6x}^{2} + 6x - 7 - {15x}^{3} + {3x}^{2} - 7x + 8$

$\tt \: (9 {x}^{3} - {15x}^{3} ) + ( { - 6x}^{2} + {3x}^{2} ) + (6x - 7x) + ( - 7 + 8)$

$\tt \: { - 6x}^{3} - {3x}^{2} - x + 1$

$:  \implies \bf \:  LHS = RHS$

$\bf \: Hence, Verified$

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$\boxed{ \purple{ \rm \: Hence, \: - {15x}^{3} + {3x}^{2} - 7x + 8 \: must \: be \: added}}$

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