Equation - Remainder Theorem
When divided by x − 1 , the polynomial P ( x ) = x 5 + 2 x 3 + A x + 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 to − 3 1 4 . find A .
The answer is 4.
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2 solutions
Good solution. But I suggest you to use latex. Your solution will be more clear and you can get more upvotes.
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thx. But how does latex works. Do you mind show me an example?
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Ok, let us take your example.
Enter the code x^3 - 12x^2 - 42 as it is in . You will get it as x 3 − 1 2 x 2 − 4 2 .
Similarly you can write the function f(x) in the Latex Code .
For more you can see maths formating guide and latex code usage wikies and notes in the search option.
Relevant wiki: Remainder Factor Theorem - Intermediate
Given that P ( x ) = x 5 + 2 x 3 + A x + B . Using remainder factor theorem, we have
{ P ( 1 ) = 1 5 + 2 ( 1 3 ) + A ( 1 ) + B = 2 P ( − 3 ) = ( − 3 ) 5 + 2 ( − 3 ) 3 + A ( − 3 ) + B = − 3 1 4 ⟹ A + B = − 1 ⟹ − 3 A + B = − 1 7 . . . ( 1 ) . . . ( 2 )
( 1 ) − ( 2 ) : 4 A = 1 6 ⟹ A = 4
Remainder Theorem The remainder of the division of a polynomial f(x) by a linear polynomial x- r is equal to f(r) In particular, x - r is a divisor of f(x) if and only if f(r)=0
Ex: Let f(x)=x3 -12x2 - 42. Polynomial division of f(x) gives the quotient x2 - 9x - 27 and a remainder of -123. Therefore, f(3) = -123.
Solution: Because remainder = 2 when equation divided by x-1. Therefore, 15 + 2(13) + A + B = 2
Because remainder = -314 when equation divided by x+3 Therefore, (-3)5 + 2(-3)3 + A(-3) + B = -314 —> -3A + B = -17
Solve equations and get A = 4, B = -5