Question

if (
$if \: (x - 2)and(x + 3)are \: factors \: of \: x3 + ax2 + bx - 30 \: find \: a \: and \: b$
​

Answer 1

Answer:

ANSWER

Given,

f(x)=x

3

+ax

2

+bx–12

Let us assume

x−2=0 and x+3=0

⇒x=2 and x=−3

Also given that, (x−2) and (x+3) are factors of f(x)

∴ By factor theorem, f(2)=0 and f(−3)=0

Now, f(2)=0

⇒23+a(2)2+b(2)−12=0

⇒8+4a+2b−12=0

⇒4a+2b−4=0

Dividing te equation by 2

⇒2a+b−2=0...(i)

And, f(−3)=0

⇒(−3)

3

+a(−3)

2

+b(−3)−12=0

⇒−27+9a−3b−12=0

⇒9a−3b−39=0

Dividing the equation by 3

⇒3a–b=13...(ii)

Now, adding equation (i) and equation (ii) we get,

(2a+b)+(3a−b)=2+13

⇒2a+3a+b−b=15

⇒5a=15

⇒a=

5

15

=3

Substituting this value of a in equation (i)

⇒2a+b=2

⇒2(3)+b=2

⇒6+b=2

⇒b=2−6

⇒b=−4

Hence, a=3 and b=−4