if x³+ax²+bx+4 is divided by x-2, the remainder is 6. if it is divided by x+1, the remainder is -3. find a and b.
Answers
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,
a=-3 and b=-1
QUESTION:-
if x³+ax²+bx+4 is divided by x-2, the remainder is 6. if it is divided by x+1, the remainder is -3. find a and b.
EXPLANATION:-
Let,
p(x)=x³+ax²+bx+4
Let's use remainder theorem here,
x-2=0
x=2
Now acc. to the remainder theorem ,
p(2)=6 (It is given that when x-2 is divided by the polynomial then the remainder is 0)
p(2)= (2)³+a(2)²+b(2)+4
p(2)=8+4a+2b+4
p(2)=12+4a+2b
p(2)=2(6+2a+b)
Since p(2)=6,
6=2(6+2a+b)
6/2=6+2a+b
3=6+2a+b
-3=2a+b ------[EQ-1]
Now similarily we can write:-
x+1=0
x=-1
So,
p(-1)=-3 (SAME REASON)
p(-1)=(-1)³+a(-1)²+b(-1)+4
p(-1)=-1+a-b+4
p(-1)=3+a-b.
p(-1)=-3
-3=3+a-b
-6=a-b --------[EQ-2]
From eq-1 and 2 we have,
-3=2a+b
-3-2a=b
Put -3-2a=b in the 2 eq
6=a-b
6=a-(-3-2a)
6=a+3+2a
6=3a+3
3=3a
a=1
So the value of a is 1 ,now put a=1 in eq-2
-3=2a+b
-3=2+b
-3-2=b
b=-5
Thus,
a=1 and b=-5
RELATED QUESTION:-
brainly.in/question/43577807
brainly.in/question/43604252