Question

if the polynomial (2x³+ax²+3x-5) and (x³+x²-2x+a) leave the same remainder when divided by (x -2),find the value of a. Also find the remainder in each case


plz answer fast​

Answer 1

Step-by-step explanation:

let x-2=0

x=2

p(x)=(2x³+ax²+3x-5)=0

=2×2³+a×2²+3×2-5=0

=2×8+4a+6-5=0

16+4a+1=0

17+4a=0

4a= -17 (equation i)

p(x) =(x³+x²-2x+a)=0

p(2) =2×2³+2²-2×2+a=0

2×8+4-4+a=0

16+a=0

a= -16(equation ii)

(equation i) -(equation ii)

4a=-17

a= -16

- +

________

3a=-1

a= -1/3

please give me BRAINLEST

Answer 2

$\huge \rm {Answer:-}$

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$\sf \pink {Given:-}$

★The two polynomials,

$\dashrightarrow {2x^{3}+ax^{2}+3x-5}$

★And,

$\dashrightarrow {x^{3}+x^{2}-2x+a}$

★leaves the same remainder,when divided by x-2

$\dashrightarrow {x-2=0}$

$\dashrightarrow {x=2}$

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★Substituting 'x=2' in the two given polynomials :-

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$\sf \blue {Polynomial\: 1:-}$

$\dashrightarrow {P(x)=2x^{3}+ax^{2}+3x-5}$

$\dashrightarrow {P(2)=2(2)^{3}+a(2)^{2}+3(2)-5}$

$\dashrightarrow {P(2)=2\times8+a\times4+6-5}$

$\dashrightarrow {P(2)=16+4a+1}$

$\dashrightarrow {\fbox{P(2)=4a+17}}$

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$\sf \orange {Polynomial\: 2:-}$

$\dashrightarrow {Q(x)=x^{3}+x^{2}-2x+a}$

$\dashrightarrow {Q(2)=(2)^{3}+(2)^{2}-2(2)+a}$

$\dashrightarrow {Q(2)=8+\cancel4-\cancel4+a}$

$\dashrightarrow {\fbox {Q(2)=8+a}}$

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★As they leave the same remainder,the two polynomials can be equated.

$\dashrightarrow {P(2)=Q(2)}$

$\dashrightarrow {4a+17=8+a}$

★Bringing like terms (a-terms) to one side,

$\dashrightarrow {4a-a=8-17}$

$\dashrightarrow {3a=-9}$

$\large \dashrightarrow {a=\frac{-9}{3}}$

$\large \implies {\fbox {a=-3}}$

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$\sf \purple {Thus,}$

$\sf \to {The\: value\: of\: a\: is\: -3}$

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$\sf \red {Remainder\: of\: the\: polynomials-}$

$\implies {x=2\: and\: a=-3}$

★POLYNOMIAL-1:

$\dashrightarrow {2x^{3}+ax^{2}+3x-5}$

$\dashrightarrow {2(2)^{3}+(-3)(2)^{2}+3(2)-5}$

$\dashrightarrow {2\times8+(-3)\times4+6-5}$

$\dashrightarrow {16-12+1}$

$\large \implies {\fbox {5}}$

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★POLYNOMIAL-2:

$\dashrightarrow {x^{3}+x^{2}-2x+a}$

$\dashrightarrow {(2)^{3}+(2)^{2}-2(2)+(-3)}$

$\dashrightarrow {8+\cancel4-\cancel4-3}$

$\dashrightarrow {8-3}$

$\large \implies {\fbox {5}}$

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$\sf \green {Henceforth,\: Verified✓}$

★The two polynomials leave the same remainder-->5

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