Remainder Theorem / Factor Theorem

Question

Hi Math Stack Exchange Community!

I'm having a hard time approaching this question.

"The polynomial $x^4 + px^3 + qx^2 + rx + 6$ is exactly divisible by each of $x-1$, $x-2$ and $x-3$. Find the values of $p, q$ and $r$."

So far I've developed one way of approach: make the polynomial $f(x)$, and show that $f(a), f(b)$ and $f(c)$ are $0$, or something like that. I am stuck, any help would be appreciated.

Answer 1

Hint: You may get last factor quite easily, let the last root be m then $m.1.2.3=6$
$\Rightarrow$ $m=1$
And get the value of $p,q,r$ using the roots.

Answer 2

Hint:

Given a polynomial of degree $4$, $3$ real roots are given, hence the $4$th root is real, say $a.$

Then

$p(x):=$

$x^4 +px^3+ qx^2+rx +6=$

$(x-1)(x-2)(x-3)(x-a).$

$6= (-1)\cdot (-2) \cdot (-3) \cdot (-a);$

$a=1;$

Compare the coefficients of $x^3,x^2$,and $x$ to find $p,q, r.$