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Question

If the polynomial ax3+bx2-c is divisible by x2+bx+c then ab is equal to?


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Solution

Step 1: Divide ax3+bx2-c by x2+bx+c

It is given that the polynomial ax3+bx2-c is divisible by x2+bx+c.

Let, f(x)=ax3+bx2-c

And, g(x)=x2+bx+c

On dividing f(x) by g(x), we see,

x2+bx+cax-abax3+bx-cax3+abx2+acx----abx2+b-acx-c-abx2-ab2x-abc+++b-ac+ab2x+abc-c

Here, b-ac+ab2x+abc-c is the remainder.

Step 2: Calculate the value of ab

Now, according to the question, the polynomial ax3+bx2-c is divisible by x2+bx+c, i.e., the remainder is zero.

So, equating the remainder to zero, we get,

b-ac+ab2x+abc-c=0

⇒ b-ac+ab2x+ab-1c=0

Now, for the remainder to be zero, it is necessary that both the terms in the above equation are zero.

i.e., b-ac+ab2x=0 and ab-1c=0

Taking, ab-1c=0

⇒ ab-1=0

⇒ ab-1=0

⇒ ab=1

Hence, the required value of ab is 1.


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