There are elements in $\mathbb Z [t]$ such that this sum is 1

Question

In order to solve a problem I'm facing I want to prove that there are $f_1,f_2,f_3$ elements in $\mathbb Z[t]$ such that $f_1(t)\cdot(4t-4)+f_2(t)\cdot(5t)+f_3(t)\cdot(t^2-17)=1$. In another words, I would like to show this $(4t-4,5t,t^2-17)$ is an unimodular row.

Thanks in advance

Answer 1

Here is a solution. Take $f_1(t)=4t+4$, $f_2(t)=-3t$ and $f_3(t)=-1$. Then $f_1\cdot(4t-4)+f_2\cdot (5t)+f_3\cdot (t^2-17)=1$.

Answer 2

Hint $\ $ The ideal contains $\,p_2\!-p_1 =\, t\!+\!4,\,$ so it contains $\ {-}p_3\ {\rm mod}\,\ t\!+\!4\, =\, -p_3(-4) = 1.\ \ $ QED