How do I find the quotient and remainder using synthetic division?

1 Answer
Jul 7, 2016

Let's take this as an example:

f(x) = x^3 - x^2 + x - 6

To find factors, notice that 6 has factors of pm1,pm2,pm3,pm6. Pick one and try synthetic division on it, and if you pick the right one (meaning that it divides), it'll give a remainder of 0.

I pick 2, so I am assuming that x-2 divides x^3 - x^2 + x - 6. This means the result should be of the form:

color(blue)((x^3 - x^2 + x - 6)/(x-2) = q(x) + r(x)

where q(x) is the quotient and r(x) is the remainder.

For synthetic division, we only use coefficients. So, we start with:

color(white)([(color(black)(2),color(black)(|),color(black)(1),color(black)(-1),color(black)(1),color(black)(-6)),(color(black)(-),color(black)(""),color(black)("_"),color(black)("_"),color(black)("_"),color(black)("_")),(color(black)(""),color(black)(""),color(black)(""),color(black)(""),color(black)(""),color(black)(""))])

If you don't have a term, use 0. So if you had x^3 - 6, use "1 0 0 "-"6".

The general process is:

  • Bring down the first coefficient.
  • Multiply it by the test factor (in this case, 2), and store the result under the next coefficient.
  • Subtract the result from this coefficient.
  • Repeat steps 2 and 3 until you've subtracted the result from the last coefficient.

The result is one degree lower.

=> color(white)([(color(black)(2),color(black)(|),color(black)(1),color(black)(-1),color(black)(1),color(black)(-6)),(color(black)(-),color(black)(""),color(black)("_"),color(black)(ul(2)),color(black)("_"),color(black)("_")),(color(black)(""),color(black)(""),color(black)(1),color(black)(-3),color(black)(""),color(black)(""))])

" "

=> color(white)([(color(black)(2),color(black)(|),color(black)(1),color(black)(-1),color(black)(1),color(black)(-6)),(color(black)(-),color(black)(""),color(black)("_"),color(black)(ul(2)),color(black)(ul(-6)),color(black)("_")),(color(black)(""),color(black)(""),color(black)(1),color(black)(-3),color(black)(7),color(black)(""))])

" "

=> color(white)([(color(black)(2),color(black)(|),color(black)(1),color(black)(-1),color(black)(1),color(black)(-6)),(color(black)(-),color(black)(""),color(black)("_"),color(black)(ul(2)),color(black)(ul(-6)),color(black)(ul(14))),(color(black)(""),color(black)(""),color(black)(\mathbf(1)),color(black)(\mathbf(-3)),color(black)(\mathbf(7)),color(black)(\mathbf(-20)))])

So, our result is color(blue)(x^2 - 3x + 7 - 20/(x-2)) for x ne 2. What we have is:

\mathbf(q(x) = x^2 - 3x + 7)
\mathbf(r(x) = -20/(x-2))

If r(x) = 0, then q(x) is a quotient that divides f(x) = x^3 - x^2 + x - 6.