# If the polynomial ax^{3} + bx - c is divisible by x^{2} + bx + c then what is the value of 'ab'?

A polynomial is a type of expression in which the exponents of all variables should be a whole number.

## Answer: If the polynomial ax^{3} + bx - c is divisible by x^{2} + bx + c then value of ab is 1.

As ax^{3} + bx - c is divisible by x^{2} + bx +c, then the remainder is 0.

**Explanation:**

The remainder when ax^{3} + bx - c is divided by x^{2} + bx +c is 0.

Let f(x) = ax^{3} + bx - c and g(x) = x^{2} + bx + c

Now, dividing f(x) by g(x) we get, (ax^{3} + bx - c) / (x^{2} + bx + c)

We see that the quotient comes to be (ax - ab) and the remainder = (ab² – ac + b) x + c (ab – 1)

Now as the remainder is 0

⇒(ab² – ac + b) x + c (ab – 1) = 0

⇒ (ab² – ac + b) x = 0 and c (ab – 1) = 0

Since the value of x and c is not equal to 0 (Because if x = 0 and c = 0, then the divisor x^{2} + bx + c = 0, which is not defined)

Therefore, ab^{2} – ac + b = 0 and ab – 1 = 0

⇒ ab = 1

### Hence, the value of ab if ax^{3} + bx - c is divisible by x^{2} + bx + c is 1.

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