Question

How do you long divide `(x^4 + 2x^3 - x^2 - 2x + 4) /( x^2 + x - 1)`?

Answer 1

`(x^4+2x^3-x^2-2x+4)/(x^2+x-1) = x^2+x-1 + 3/(x^2+x-1)`

Explanation:

I prefer to long divide just the coefficients, which is basically the same process without writing the powers of `x` in...

Note well that if there were any 'missing' powers of `x` in the dividend or the divisor then we would have to include a `0` coefficient for them. For example, if you were dividing by `x^2-3` then the divisor would be written as `1color(white)(x)0color(white)(x)-3` and not simply `1color(white)(x)-3`

The process is similar to long division of numbers:

Write the dividend under the bar and the divisor to the left.

Choose the first term `color(blue)(1)` of the quotient to match the leading term of the dividend when multiplied by the divisor.

Write the product below the divisor and subtract.

Bring down the next term of the dividend alongside the difference and choose the next term `color(blue)(1)` of the quotient to match the leading term of the running remainder.

Repeat until the running remainder is shorter than the divisor.

We find that:

`(x^4+2x^3-x^2-2x+4)/(x^2+x-1) = x^2+x-1 + 3/(x^2+x-1)`