Things you might have noticed

What do you notice about the two problems below?

\[\frac{6x^3+13x^2+9x+2}{3x+2}= \ldots\]

  • How can you use the multiplication grid on the right-hand side to find the result of dividing \(6x^3+13x^2+9x+2\) by \(3x+2\)?

First let’s think about the fraction on the left-hand side.

What type of result do you expect if this division is carried out?

The multiplication grid on the right-hand side also features the expression \(3x+2\) but is set up to have a quadratic expression along the top.

What type of polynomial do you expect to find inside the multiplication grid?

If we think about the division and the multiplication grid simultaneously, we can begin to fill in some of the missing values. In the grid below we suggest the order in which you might choose to complete values.

\(\color{blue}{(1)}\) \(\color{blue}{(4)}\) \(\color{blue}{(7)}\)
\(\color{blue}{(3)}\) \(\color{blue}{(6)}\) \(\color{blue}{(9)}\)
  • Why might we complete the value in the top left box of the grid first?

  • In each case try to justify why that value could be found next and try to fill it in.

  • Which values do you find more tricky to complete? Why?
\(\color{red}{6x^3}\) \(\color{red}{9x^2}\) \(\color{red}{3x}\)
\(\color{red}{4x^2}\) \(\color{red}{6x}\) \(\color{red}{2}\)

Can you now write down any number statements that this grid might represent?

How do we now bring all of our thinking together to complete the division \(\dfrac{6x^3+13x^2+9x+2}{3x+2}=\ldots\)?

What stays the same and what changes, if the division on the left-hand side is \(\dfrac{6x^3+13x^2+9x+5}{3x+2}=\ldots\) instead?

If we compare this fraction to the one considered previously we notice that the only difference is the constant term in the cubic expression, which is now \(5\) instead of \(2\). If we were to set up a multiplication grid to tackle this problem it might still look like the one above.

  • Which part of the multiplication grid tells us about the constant term in the cubic expression?

  • At what stage did we choose to complete this box in the previous problem? Do we have any choice about the order in which it is completed?

  • What does this tell us about this particular division calculation?
\(\color{red}{6x^3}\) \(\color{red}{9x^2}\) \(\color{red}{3x}\)
\(\color{red}{4x^2}\) \(\color{red}{6x}\) \(\color{red}{2}\)

We need the constant term to be \(5\). We can’t get \(5\) from the grid and still make all of the other terms work. This means that \(6x^3+13x^2+9x+5\) does not divide exactly by \(3x+2\), so there must be a remainder.

Could we have checked whether or not \(3x+2\) was a factor of \(6x^3+13x^2+9x+5\) before we began?

We can now complete the division calculation, \[\frac{6x^3+13x^2+9x+5}{3x+2}=2x^2+3x+1+ \frac{3}{3x+2}.\]

Where did the final term, \(\dfrac{3}{3x+2}\) come from?

Can you use a similar approach to divide \(4x^4+3x^3+2x+1\) by \(x^2+x+2\)? Are you convinced that taking this approach gives you the same result as other methods?

We could set up the multiplication grid like this:

  • Why is this multiplication grid larger than in the previous problems?

  • How did you know the type of expression that could go along the top of the grid?

  • Is the order in which you might now complete the grid similar to that in the previous problems?
\(\color{red}{4x^4}\) \(\color{red}{-x^3}\) \(\color{red}{-7x^2}\)
\(\color{red}{4x^3}\) \(\color{red}{-x^2}\) \(\color{red}{-7x}\)
\(\color{red}{8x^2}\) \(\color{red}{-2x}\) \(\color{red}{-14}\)

Can you now complete the division, \[\frac{4x^4+3x^3+2x+1}{x^2+x+2}=\ldots?\]

What other methods do you know for carrying out algebraic division? How do they compare to the approach taken here?