if x³ - 2 x² + a x + b has a factor( x +2) 2and leaves a remainder 9 when divided by (x+ 1 )find the values of a and b*
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Answers
Answer:
a = -4
b = 8
Note:
• Remainder theorem : If a polynomial is divided by (x - a) , then the remainder is given as : Remainder , R = p(a).
• Factor theorem : (i) If (x - a) is a factor of any polynomial p(x) , then the remainder is zero when p(x) is divided by (x - a) , ie ; p(a) = 0.
(ii) If the remainder obtained on dividing a polynomial p(x) by (x - a) is zero , ie ; if p(a) = 0 , then (x - a) is a factor of polynomial p(x) .
Solution:
Let the given polynomial be p(x) .
Thus,
p(x) = x³ - 2x² + ax + b
It is given that, (x + 2) is a factor of the given polynomial , thus ;
=> p(-2) = 0
=> (-2)³ - 2(-2)² + a(-2) + b = 0
=> - 8 - 8 - 2a + b = 0
=> -16 - 2a + b = 0
=> b = 2a + 16 -------(1)
Also,
When p(x) is divided by (x + 1) , then the remainder is 9 , thus ;
=> p(-1) = 9
=> (-1)³ - 2(-1)² + a(-1) + b = 9
=> - 1 - 2 - a + b = 9
=> -3 - a + b = 9
=> b = 9 + 3 + a
=> b = 12 + a -------(2)
From eq-(1) and (2) , we have ;
=> 2a + 16 = 12 + a
=> 2a - a = 12 - 16
=> a = - 4
Now,
Putting a = -4 in eq-(2) , we get ;
=> b = 12 + a
=> b = 12 - 4
=> b = 8
Hence,
The required values of a and b are (-4) and 8 respectively .