How do you long divide (x^3-27)/(x-3)x327x3?

1 Answer
Aug 24, 2017

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 xx.

  • 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.