Question

How do you long divide `(x^3-27)/(x-3)`?

Answer 1

It is a similar process to long division of numbers...

Explanation:

Long division of polynomials is similar to long division of numbers.

• Write the dividend under the bar and the divisor to the left, including all powers of `x`.

• Write the first term of the quotient above the bar, chosen so that when multiplied by the divisor it matches the first term of the dividend.

`color(white)(x - 30"|")underline(color(white)(0)x^2color(white)(+0x^2+0x-270)`
`x - 3color(white)(0)"|"color(white)(0)x^3+0x^2+0x-27`

• Write the product of the first term of the quotient and the divisor under the dividend, subtract it and bring down the next term of the dividend alongside the remainder.

`color(white)(x - 30"|")underline(color(white)(0)x^2color(white)(+0x^2+0x-270)`
`x - 3color(white)(0)"|"color(white)(0)x^3+0x^2+0x-27`
`color(white)(x - 30"|"0)underline(x^3-3x^2)`
`color(white)(x - 30"|"0x^3-)3x^2+0x`

• Choose the next term of the quotient so when multiplied by the divisor matches the leading term `3x^2` of our running remainder.

`color(white)(x - 30"|")underline(color(white)(0)x^2+3xcolor(white)(+0x^2-270)`
`x - 3color(white)(0)"|"color(white)(0)x^3+0x^2+0x-27`
`color(white)(x - 30"|"0)underline(x^3-3x^2)`
`color(white)(x - 30"|"0x^3-)3x^2+0x`

• Write the product of this second term of the quotient and the divisor under the remainder, subtract it and bring down the next term of the dividend alongside it.

`color(white)(x - 30"|")underline(color(white)(0)x^2+3xcolor(white)(+0x^2-270)`
`x - 3color(white)(0)"|"color(white)(0)x^3+0x^2+0x-27`
`color(white)(x - 30"|"0)underline(x^3-3x^2)`
`color(white)(x - 30"|"0x^3-)3x^2+0x`
`color(white)(x - 30"|"0x^3+)underline(3x^2-9x`
`color(white)(x - 30"|"0x^3-3x^2+)9x-27`

• Choose the next term of the quotient so when multiplied by the divisor matches the leading term `9x` of our running remainder.

`color(white)(x - 30"|")underline(color(white)(0)x^2+3xcolor(white)(.)+9color(white)(-270x)`
`x - 3color(white)(0)"|"color(white)(0)x^3+0x^2+0x-27`
`color(white)(x - 30"|"0)underline(x^3-3x^2)`
`color(white)(x - 30"|"0x^3-)3x^2+0x`
`color(white)(x - 30"|"0x^3+)underline(3x^2-9x`
`color(white)(x - 30"|"0x^3-3x^2+)9x-27`

• Write the product of this third term of the quotient and the divisor under the remainder and subtract it.

`color(white)(x - 30"|")underline(color(white)(0)x^2+3xcolor(white)(.)+9color(white)(-270x)`
`x - 3color(white)(0)"|"color(white)(0)x^3+0x^2+0x-27`
`color(white)(x - 30"|"0)underline(x^3-3x^2)`
`color(white)(x - 30"|"0x^3-)3x^2+0x`
`color(white)(x - 30"|"0x^3+)underline(3x^2-9x`
`color(white)(x - 30"|"0x^3-3x^2+)9x-27`
`color(white)(x - 30"|"0x^3-3x^2+)underline(9x-27)`

• In this example, there is no remainder. The division is exact. If we did get a remainder with degree less than the divisor then we would stop here anyway.