How do you do the following divisions and find quotient and remainder a3-b3 by a-b?

Answer 1

0

Answer 2

You can use the remainder theorem to show that the remainder on division is 0. If you substitute a = b in a3 - b3 the value is 0. Therefore (a-b) is a factor of a3 - b3 and so the remainder is 0.

To find the quotient, you do the division like you'd do a long division with numbers. It is a bit difficult to work with this browser, but here goes:

a goes into a3: a2 times so the first term of the quotient is a2.

Now a2*(a-b) = a3 - a2b

Subtract this expression from a3 - b3

So the remainder, at this stage, is a2b - b3

Now a goes into a2b: ab times so the next term of the quotient is ab.

ab*(a-b) = a2b - ab2

Subtract this from the last remainder to give the new remainder of ab2 - b3

A goes into ab2: b2 times so the next term of the quotient is b2.

b2*(a-b) = ab2 - b3

Subtract this from the previous remainder to give the next remainder of 0.

Done.

So, a3 - b3 = (a - b)*(a2 + ab + b2)