Question

Fo the equation `5x^5 + 13x^4 - 2x^2 -6, x-1` , How to divide the first expression by the second?

Can someone please give me a shortcut or a explanation/methods I can apply for similar complicated question like this one (if possible). Thank you.

Answer 1

`(5x^5+13x^4-2x^2-6)/(x-1) = 5x^4+18x^3+18x^2+16x+16+10/(x-1)`

Explanation:

One method is to redistribute the terms in the dividend into multiples of the divisor, which you can do like this:

`5x^5+13x^4-2x^2-6`

`=5x^5-5x^4+18x^4-2x^2-6`

`=5x^5-5x^4+18x^4-18x^3+18x^3-2x^2-6`

`=5x^5-5x^4+18x^4-18x^3+18x^3-18x^2+16x^2-6`

`=5x^5-5x^4+18x^4-18x^3+18x^3-18x^2+16x^2-16x+16x-6`

`=5x^5-5x^4+18x^4-18x^3+18x^3-18x^2+16x^2-16x+16x-16+10`

`=(x-1)(5x^4+18x^3+18x^2+16x+16)+10`

So:

`(5x^5+13x^4-2x^2-6)/(x-1) = 5x^4+18x^3+18x^2+16x+16+10/(x-1)`

Alternative method

The way I would do the division would be to start writing the factorisation and add terms one at a time...

The first term of the quotient must be `5x^4` in order that when multiplied by `x` it gives the leading term `5x^5`, so start to write down:

`5x^5+13x^4-2x^2-6 = (x-1)(5x^4...`

Then note that `(-1)(5x^4)` will give us `-5x^4`, but we want `13x^4`. So we can deduce that the next term in the quotient is `18x^3`:

`5x^5+13x^4-2x^2-6 = (x-1)(5x^4+18x^3...`

This will result in a term `-18x^3` which we don't want, so the next term in our quotient is `18x^2`...

`5x^5+13x^4-2x^2-6 = (x-1)(5x^4+18x^3+18x^2...`

This will result in a term `-18x^2`, but we want `-2x^2`. So the next term we need for our quotient is `16x`...

`5x^5+13x^4-2x^2-6 = (x-1)(5x^4+18x^3+18x^2+16x...`

This will give result in `-16x`, but there is no `x` term in the dividend, so we need a constant term `16` to cancel it out...

`5x^5+13x^4-2x^2-6 = (x-1)(5x^4+18x^3+18x^2+16x+16)...`

Finally, note that `(-1)(16) = -16`, but we want `-6`, so we need to add a remainder term `10`...

`5x^5+13x^4-2x^2-6 = (x-1)(5x^4+18x^3+18x^2+16x+16)+10`

This takes a lot longer to describe than to do. With a bit of practice you may find this the most convenient method.

Answer 2

Quotient `5x^4 +18x^3 +18x^2 +16x` Remainder 10.

Explanation:

Shortcut method is synthetic division. To do this, write the coefficients of all the terms, including the missing ones, starting with the highest degree term in a single row. The missing terms here are `x^3` and `x`. Their coefficients would be o s. To the left of this row write the value of x ,obtained by equating the divisor to 0. In this case it would be x=1

            1|   5         13           0         -2       0           -6

 Now divide the first number 5 by 1 and the write the quotient 5 below the next number 13.

Add them to get 18.

Now divide this number by 1 and write the quotient 18 below the next number 0. continue this process till the last digit. The final display would be as follows

            1|   5         13           0          -2      0            -6
                            _5______18____18___16_____16___________
                             18           18        16     16          10

The Last digit obtained is 10. This is the Remainder. The quotient would be `5x^4 +18x^3 +18x^2 +16x +16`