x³-ax²+bx+7 and ax³+7x²-5x+b leaves the remainder -1 and 7 on division by x+1 and x-2 respectively, find a and b
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Answer:
The value of a and b are -3 and -1 respectively.
Step-by-step explanation:
Let p(x)=x³+ax²+bx+6
i ) it is given that , (x-2) is a factor of p(x) ,then
p(x)=0
=> 2³+a(2)²+b×2+6=0
=> 8+4a+2b+6=0
=> 4a+2b+14=0
Divide each term by 2 , we get
=> 2a+b+7=0 ---(1)
ii)It is given that, if p(x) divided by (x-3) leaves a remainder 3
=> p(3)=3
=> 3³+a(3)²+b×3+6=3
=> 27+9a+3b+6-3=0
=> 9a+3b+30=0
Divide each term by 3 , we get
=> 3a+b+10=0 ---(2)
Subtract equation (1) from equation (2) , we get
a + 3 = 0
=> a = -3
Now,
Substitute a=-3 in equation (1) , we get
2(-3)+b+7=0
=> -6+b+7=0
=> b+1=0
=> b = -1
Therefore,
The value of a and b are -3 and -1 respectively.
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