Question

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 1

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.