# the value of a determinant

Let be a polynomial function $$P\in \mathbb{R}[X]$$.If I divide $$P$$ by $$(x-1)(x-2)(x-3)(x-4)$$ I get a remainder without "free term" ( like $$ax^{3}+bx^{2}+cx$$ )

I have to calculate the determinant:

$$\begin{vmatrix} P(1) & 1 & 1 &1 \\ P(2) & 2 & 4 &8 \\ P(3) & 3 & 9 &27 \\ P(4) & 4 & 16 &64 \end{vmatrix}$$

My try: I wrote that $$P(x)=(x-1)(x-2)(x-3)(x-4)\cdot Q(x)+ax^{3}+bx^{2}+cx$$

So for $$x=1=> P(1)=a+b+c$$

$$x=2=> P(2)=8a+4b+2c$$

$$x=3=> P(3)=27a+9b+3c$$

$$x=4=> P(4)=64a+16b+4c$$

And now I just have to replace the results in my determinant but it takes me a lot of time to solve the determinant.I'm wondering if there is a short way to solve this. Can you help me with some ideas?

Hint: Take column one and subtract $$c$$ from column $$2,$$ $$b$$ from column 3 and $$a$$ of column 4. Do you get a column full of $$0's$$? What is the determinant of that?
• Your method is fine but multiply the second column by $c$ instead and the third one by $b$ and the fourth one by $a.$ You should get 0. And determinant is never changed by those operations so determinant should be 0. – Phicar Apr 15 at 19:59
• yes. A vector full of $0's$ gives you determinant $0$ – Phicar Apr 15 at 20:24
Hint: What is $$a$$ times the fourth column, plus $$b$$ times the third column, plus $$c$$ times the second?