Is there a quicker way to calculate the square of quadratic polynomial?

Question

For example, consider

$$ (x^2 - 6 x + 7)^2 $$

Apart from

$$ x^2 (x^2-6x+7) - 6x (x^2-6x+7)+ 7 (x^2-6x+7) $$

are there any quicker ways of calculation the square of $x^2 - 6 x + 7$?

Answer 1

In your mind visualise a three column by three row answer grid.

The column headings are $x^2$,$-6x$ and $7$, the row headings the same. Write down the bits as you visualise the row, column intersections. On the leading diagonal will be $x^4$, $36x^2$ and $49$

Each of the other answers occurs twice. So, essentially, you are trying to visualise the following.

$$\begin{bmatrix}x^2 & -6x &7 \end{bmatrix}$$ $$\begin{bmatrix}x^2 \\ -6x \\7 \end{bmatrix}\begin{bmatrix}x^4 & -6x^3 &7x^2 \\-6x^3 & 36x^2 & -42x\\7x^2 &-42x &49\end{bmatrix}$$

With practice you should be able to simply write down the "almost answer", $$x^4+36x^2+49-12x^3+14x^2-84x$$ and it depends on how much you can carry in your head as to if you want to gather together like terms at the same time. Such mental gymnastics helps to keep the algebra interesting.

The party piece is to do similar with $(x^2-7x+6)^3$ by visualising a cube!

Here is a link to a diagram of the technique as a "visual proof" (on FaceBook) : https://www.facebook.com/examath/photos/a.166108407455452/541471543252468/