Largest number n

Find the largest natural number d that has that property for any natural number n; the number V(n) is the value of the expression:

V (n) = n ^ 4 + 11n²−12

is divisible by d.

Final Answer:

d =  12

Step-by-step explanation:

$$\begin{gathered} V(n)=n^4+11n^2-12 \\ V(n)=n^2(n^2+11)-12 \\ t=n^2 \\ {t}^2+11t-12=0 \\ {t}^2+11t-12=0 \\ a=1;b=11;c=-12 \\ D=b^2-4ac=11^2-4\cdot 1\cdot (-12)=169 \\ D>0 \\ {t}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-11\pm \sqrt{169}}{2} \\ {t}_{1,2}=\frac{-11\pm 13}{2} \\ {t}_{1,2}=-5.5\pm 6.5 \\ {t}_1=1 \\ {t}_2=-12 \\ V(n)=(n^2-1)(n^2+12)=(n-1)(n+1)(n^2+12) \\ D=\{1,2,3,4,6,12\} \\ d=12 \end{gathered}$$

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