Question

If x² + x - 12 divides p(x) = x³ + ax² + bx - 8 exactly, find a and b​

Answer 1

Answer:

x  

2

+x−12=0⇒x  

2

+4x−3x−12=0⇒(x−3)(x+4)=0  

so x=3 or −4

p(x)÷g(x)=q(x)+r⇒p(x)÷g(x)=q(x)+0

x  

3

+ax  

2

+bx−84÷x  

2

+x−12

after dividing remainder will be ax  

2

−x  

2

+bx+12x−84=0

Let x=3⇒27a−9+3b+36−84=0

9a+3b=57⇒3(3a+b)=57

3a+b=19...(1)

Let x=−4

16a−16−4b−48−84=0⇒16a−4b=148

4(4a−b)=148⇒4a−b=37...(2)

Add (1) & (2)

3a+b+4a−b=19+37

7a=56

a=8

Now a=8 in (1)

3a+b=19

24+b=19

b=−5

Answer 2

Question :-

If x² + x - 12 divides f(x) = x³ + ax² + bx - 84 exactly. find a and b.Given

The equation  x² + x - 12 divides the equation x³ + ax² + bx - 84 exactly.

To Find :-

Value of a and b.

Solution :-

The equation  x² + x - 12 divides the equation x³ + ax² + bx - 84 exactly.

Therefore, we have the remainder equals to 0.

On Dividing, the given equation by x² + x - 12, we get  

(a - 8)x² + (b + 5)x = 0

On comparing the equation with the given equation, we get

a = 8 and b = -5

Hence, the values of a and b are 8 and -5.

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