14. Find the values of a and b so that the polynomial ( x4 +ax3 - 7x2 -8x + b) is exactly divisible by (x+2) as well as (x+3)
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Answer:
The values of a and b are 38/21 and 220/21 respectively.
Given:
- p(x)=x⁴+ax³-7x²-8x+b is exactly divisible by (x+2) and (x+3).
To find:
- The value of a and b.
Solution:
If p(x) is exactly divisible by (x+2) as well as (x+3) then by Remainder theorem we get
p(-2)=0 as well as p(-3)=0
p(x)=x⁴+ax³-7x²-8x+b
p(-2)=(-2)⁴+a(-2)³-7(-2)²-8(-2)+b
>> 0=16+-8a-28+16+b
>> -8a+b=-4...(1)
p(-3)=(-3)⁴+a(-3)³-7(-3)²-8(-3)+b
>> 0=81-27a-63+24+b
>> -27a+b=-42...(2)
Subtract eq(2) from eq(1), we get
⠀⠀⠀⠀-8a+b=-4
⠀⠀⠀-
⠀⠀⠀⠀-27a+b=-42
___________________
⠀⠀⠀⠀21a=38
⠀⠀⠀⠀⠀a=38/21
Substitute a=38/21 in eq(1), we get
>> -8(38/21)+b=-4
>> b=-4+304/21
>> b=(-84+304)/21
>> b=220/21
The values of a and b are 38/21 and 220/21 respectively.
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