When divided by x - 1, the polynomial P(x) = x5 + 2x3 +Ax + B, where A and B are constants, the remainder is equal to 2. When P(x) is divided by x + 3, the remainder is equal -314. Find A and B.
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Answers
Answer:-
Given:-
When P(x) = x⁵ + 2x³ + Ax + B is divided by x - 1 ; the remainder obtained is 2.
So, the divisor ,
g(x) = x - 1
⟹ g(x) = 0
⟹ x - 1 = 0
⟹ x = 1
Simply, Substitute x = 1 in the given polynomial.
⟹ P(1) = 1⁵ + 2(1)³ + A(1) + B
The remainder is 2.
So,
⟹ 2 = 1 + 2 + A + B
⟹ 2 - 2 - 1 = A + B
⟹ - 1 = A + B -- equation (1)
And also given that,
when it is divided by x + 3 , the remainder obtained is - 314.
Here,
g(x) = x + 3
⟹ x + 3 = 0
⟹ x = - 3
Substitute x = - 3 in the same polynomial.
⟹ P( - 3) = ( - 3)⁵ + 2 ( - 3)³ + A ( - 3) + B
⟹ - 314 = - 243 - 54 - 3A + B
⟹ - 314 + 243 + 54 = - 3A + B
⟹ - 17 = - 3A + B -- (2)
Subtract equation (1) from (2).
⟹ - 3A + B - (A + B) = - 17 - ( - 1)
⟹ - 3A + B - A - B = - 17 + 1
⟹ - 4A = - 16
⟹ A = - 16 / - 4
⟹ A = 4
Substitute the value of y in equation (1).
⟹ A + B = - 1
⟹ 4 + B = - 1
⟹ B = - 1 - 4
⟹ B = - 5
∴ The values of A and B are 4 & - 5 ; respectively.
..QUESTION..ㅤㅤ
When divided by x - 1, the polynomial P(x) = x⁵ + 2x³ +Ax + B, where A and B are constants, the remainder is equal to 2. When P(x) is divided by x + 3, the remainder is equal - 314. Find A and B.
..ANSWER..
Given ,
ㅤㅤㅤ➪ P ( x ) = x⁵ + 2x³ + Ax + B
ㅤㅤㅤ➪ g ( x ) = x - 1
ㅤㅤㅤnote* » Remainder = 2ㅤㅤ [ 1 ]
•°• Zeroes of g ( x ) »
ㅤㅤㅤ➪ x - 1 = 0
ㅤㅤㅤ➪ x = 0 + 1
ㅤ ㅤ ➪ x = 1ㅤ
ㅤㅤㅤㅤㅤ
•°• Putting values »
ㅤ➪ P ( x ) = x⁵ + 2x³ + Ax + B
ㅤ➪ P ( 1 ) = ( 1 )⁵ + 2 ( 1 )³ + A ( 1 ) + B
ㅤㅤㅤㅤ ㅤ= 1 + 2 + A + Bㅤㅤㅤ[ 2 ]
Now ,
ㅤ➪ 2 = 1 + 2 + A + B [ combining (1) , (2) ]
ㅤ➪ 1 + 2 + A + B = 2
ㅤ➪ 2 + A + B = 2 - 1
ㅤ➪ A + B = 1 - 2
ㅤ➪ A + B = ( - 1 )ㅤㅤㅤ----------- [ (3) ]
☯︎ Also , given here
When the P(x) is divided by g(x) x + 3 , the result/remainder is = ( - 314 )ㅤㅤ[ Given ★ ]
•°• Zeroes of the g ( x ) »
ㅤㅤㅤ➪ g ( x ) = 0
ㅤㅤㅤ➪ x + 3 = 0
ㅤㅤㅤ➪ x = 0 - 3
ㅤㅤㅤ➪ x = ( - 3 )
☯︎ Putting values in the P ( x ) , we get »
➪ P (x) = x⁵ + 2x³ + Ax + B
➪ P ( - 3 ) = ( -3 )⁵ + 2 ( -3 )³ + A(-3) + B
➪ㅤㅤㅤㅤ= - 243 + 2 ( -27 ) + ( -3A ) + B
➪ㅤㅤㅤㅤ= - 243 - 54 - 3A + B
➪ - 314 ㅤ= - 297 - 3A + B [From (given ★)]
➪ - 314 + 297 = - 3A + B
➪ - 17 = - 3A + B
➪ -3A + B = - 17ㅤㅤㅤ----------- [ (4) ]
☯︎ Now [ (4) - (3) ] ,
➪ - 3A + B - ( A + B ) = - 17 - ( - 1 )
➪ - 3A + B - A - B = - 17 + 1
➪ - 4A = - 16
➪ A = ( - 16 ) / ( - 4 )
➪ A = 4ㅤㅤㅤㅤㅤㅤ---------- ( 5 )
ㅤㅤㅤ☯︎ Again , B =
➪ - 3A + B = - 17
➪ - 3 ( 4 ) + B = - 17 [ From ( 5 ) ]
➪ - 12 + B = - 17
➪ B = - 17 + 12
➪ B = - 5
•°• ㅤㅤㅤValue of A = 4
ㅤㅤㅤㅤ Value of B = - 5