Question

the polynomial x cube + 2 x square - 5 x minus 8 and x cube + x square - 12 x minus 6 when divided by (x minus 2) and (x minus 3) leave reminders P and Q respectively if Q minus b is equal to 10 find the value of a​

Answer 1

Solution==>
Let,
f(x)=x³+2x²-5ax-8
g(x)=x³+ax²-12x-6

According to question, when f(x) and g(x) is divisible by (x-2) and (x-3) leave remainder P and Q respectively.

Now,
when f(x) is divisible by (x-2)
------------------------------------------
------------------------------------------
f(x)=x³+2x²-5ax-8
x-2=0
x=2
put x=2 in f(x)
f(2)=(2)³+2(2)²-5a(2)-8
f(2)=8+8-10a-8
f(2)=8-10a
Remainder P=8-10a

also,
when g(x) is divided by (x-3)
-----------------------------------------
-----------------------------------------
g(x)=x³+ax²-12x-6
x-3=0
x=3
now, put x=3 in g(x)
g(3)=(3)³+a(3)²-12(3)-6
g(3)=27+9a-36-6
g(3)=27+9a-42
g(3)=9a-15
Remainder Q=9a-15

ALSO Given, remainders (Q-P) are equal to 10
Q-P=10
9a-15-(8-10a)=10
9a-15-8+10a=10
19a=10+15+8
19a=33
a=33/19
Answer....

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