Question

find the value of a and b so that the polynomial x3 + 10x2 + ax + b is exctally divisible by x-1 and x-2

Answer 1

polynomial , x³ +10x² + ax + b is exactly divisible by ( x -1) and (x -2)

it means x = 1 and 2 are the zeors of given polynomial .

so,
put x = 1

(1)³ + 10(1)² +a(1) + b = 0

1 + 10 + a + b = 0

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

again put x =2

(2)³ +10(2)² +a(2) + b =0

8 + 40 + 2a + b = 0

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

solve equations (1) and (2)

a = -37

b = 26

Answer 2

let given polynomial be,
f(x)=x³+10x²+ax+b
also given that f(x) is exactly divisible by (x-1) and (x-2)
Therefore 1 and 2 are zeroes of given polynomial f(x).
Now we have,
f(x)=0
f(1)=0
(1)³+10(1)²+a(1)+b=0

1+10+a+b=0

a+b=-11

b=-11-a......................................1
and
f(2)=0
(2)³+10(2)²+a(2)+b=0

8+40+2a+b=0

48+2a+b=0.............................2
By putting eq1 in eq2 we get,
48+2a-11-a=0

a=-37
Putting value of a in eq1 we get,
b=-11-(-37)=-11+37=26


Hence value of a=-37 and b=26.