Question

Find the values of a and b so that y^3 + 10y^2 + ay + b is exactly divisible by y - 1 and y - 2

Answer 1

Answer:

a=-37  and b=26

Step-by-step explanation:

Concept used:

Factor theorem:

(x-a) is a factor of f(x) iff f(a) =0

Let,

$f(y)=y^3+10y^2+ay+b$

since (y-1) is a factor of f(y), f(1)=0

$\implies\:1^3+10(1)^2+a(1)+b=0$

$\implies\:1+10+a+b=0$

$\implies\:a+b=-11$........(1)

since (y-2) is a factor of f(y), f(2)=0

$\implies\:2^3+10(2)^2+a(2)+b=0$

$\implies\:8+40+2a+b=0$

$\implies\:2a+b=-48$........(2)

Now, we have

$a+b=-11$........(1)

$2a+b=-48$........(2)

(1)-(2) $\implies$

-a=37

a=-37

put a=-37 in (1) we get

-37+b=-11

b=26