How Do I Solve This Linear Algebra Equation?

[Sandra_xx]

Hi!

This is my first post at this forum. At the look of it I think i will be spending a lot of time here. I´ve have a problem that i need help with. Me and my boyfriend started a discussion yeasterday who was the smartest of us. So we gave each other a assignment. Mine was:

Solve the equation

| 1 1 1 1|
| x 2 3 4|
| x^2 4 9 16| = 0
| x^3 8 27 64|

I´ve have thought out that i need to use linear algebra to solve this. This is were my problem kick in. I´ve have just read moderate linaer algebra. So please can you help me. Don´t want to be the dumb one We didn´t say how we should solve it...

Thanks in advance! /X Sandra

 

[HallsofIvy]

Expand the determinant using the first column:
$$\left|\begin{array}{cccc}1 & 1 & 1 & 1 \\ x & 2 & 3 & 4 \\ x^2 & 4 & 9 & 16 \\ x^3 & 8 & 27 & 65 \end{array}\right|=$$
$$\left|\begin{array}{ccc}2 & 3 & 4 \\ 4 & 9 & 16 \\ 8 & 27 & 64\end{array}\right|$$
$$- x \left|\begin{array}{ccc}1 & 1 & 1 \\ 4 & 9 & 16 \\ 8 & 27 & 64\end{array}\right|$$
$$+ x^2 \left|\begin{array}{ccc}1 & 1 & 1 \\2 & 3 & 4 \\ 8 & 27 & 64\end{array}\right|$$
$$- x^3 \left|\begin{array}{ccc}1 & 1 & 1 \\2 & 3 & 4 \\ 4 & 9 & 16\end{array}\right|$$

Which gives a single cubic (not linear) equation to solve. (Hint: it has one integer solution.)

 

[Sandra_xx]

Got it now. Thank you :!)

X/ Sandra

 

[HallsofIvy]

And give your boyfriend a poke in the eye with a sharp stick!