Question

How do you long divide `(3x^5 + 8x^4 - 2x^3 + x^2 +3x + 6) / (3x^2 + x +4)`?

Answer 1

`(3x^5+8x^4-2x^3+x^2+3x+6)/(3x^2+x+4)`

`=x^3+7/3x^2-25/9x-50/27+(431/27x+362/27)/(3x^2+x+4)`

Explanation:

You can long divide these polynomials like this...

The process is similar to long division of numbers.

Write the dividend (`3x^5+8x^4-2x^3+x^2+3x+6`) under the bar and the divisor (`3x^2+x+4`) to the left of the bar.

Choose the first term `color(blue)(x^3)` of the quotient so that when multiplied by the divisor it matches the first term (`3x^5`) of the dividend.

Write the product of this first term of the quotient and the divisor under the dividend. and subtract it to give a running remainder.

Bring down the next term `x^2` of the dividend alongside it.

Choose the next term `color(blue)(7/3x^2)` of the quotient to match the leading term of the running remainder when multiplied by the divisor.

Write the product of this term and the divisor under the running remainder and subtract it, etc.

Repeat until the running remainder is shorter than the divisor.

In this example we find that the quotient is:

`x^3+7/3x^2-25/9x-50/27`

with remainder:

`431/27x+362/27`