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(A) To find the number of prizes left undistributed,we need to divide the total number of prizes $P(x) = 3x^3 + 10x^2 + 7x - 2$ by the number of stude

Vedclass · Accepted Answer

$0$

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(A) To find the number of prizes left undistributed,we need to divide the total number of prizes $P(x) = 3x^3 + 10x^2 + 7x - 2$ by the number of students $(x + 2)$.Using polynomial long division:Step $1$: Divide $3x^3$ by $x$ to get $3x^2$.Step $2$: Multiply $3x^2(x + 2) = 3x^3 + 6x^2$.Step $3$: Subtract $(3x^3 + 10x^2 + 7x - 2) - (3x^3 + 6x^2) = 4x^2 + 7x - 2$.Step $4$: Divide $4x^2$ by $x$ to get $4x$.Step $5$: Multiply $4x(x + 2) = 4x^2 + 8x$.Step $6$: Subtract $(4x^2 + 7x - 2) - (4x^2 + 8x) = -x - 2$.Step $7$: Divide $-x$ by $x$ to get $-1$.Step $8$: Multiply $-1(x + 2) = -x - 2$.Step $9$: Subtract $(-x - 2) - (-x - 2) = 0$.The remainder is $0$. Therefore,no prizes are left undistributed.

In a prize-winning competition,$x+2$ students are qualified for the prize. If a total of $3x^3 + 10x^2 + 7x - 2$ prizes are distributed among them equally,find the number of prizes left undistributed,where $x \in N$.

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