partial fractions. pls answer my question. don't spam.
Answers
Question :
The remainders of the polynomial f(x) when divided by x+1, x+2, and x-2 are 6, 15 and 3 respectively .
The remainder of f(x) when divided by (x+1)(x+2)(x-2) is :
Answer :
{ a } , 2x² - 3x + 1
Solution :
The given polynomial in question is a quadratic polynomial .
This can be written as :
f ( x ) = ( x + 1) Q1x + 5 = (x + 2) Q2x + 15 = ( x - 2) Q2x + 3
Let us find the value of x = 2
=> f ( 2 ) = 3 × [ Q1 x ] + 5 = 4 × [ Q2 x ] + 15 = 0 + 3
=> f ( 2) = 3
[> Dealing each cases individually :
3 Q1 x + 5 = 3
3 Q1( x) = -2
=> Remainder obtained when divided by Q1(x) is -2/3
[>
4 × [ Q2 x] + 15 = 3
=> 4 [ Q 2 ] x = -12
=> Q2(x) = -3
=> Remainder obtained when divided by Q 2(x) is -3
Now , we can similarly find the value of Q 3(x) by placing the value of x as -1 in f (x)
Then by using the vieta ' s summation of roots formulas , we will obtain the value of f(x) as :
=> x³ + 3x² - 7x - 3
=> x³ + x² - 4x - 4 + 2x² - 3x + 1
=> ( x + 1)( x + 2)( x - 2) + 2x^2 - 3x + 1
Hence , option A is the correct answer .
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