I cannot tell you a general approach to solving quintics, but the integer

root theorem lets you solve this one with ease.

[tex]x^5 + 3x + 6 = 10 \iff x^5 + 3x - 4 = 0.[/tex]

Now, by the integer root theorem, if there is a rational root, it will be an integer that divides 4 evenly. There are only six possibilities, - 4, - 2, - 1, 1, 2, and 4. Obviously, a negative number will not work. Just as obviously, 2 and 4 won't work. How about + 1.

[tex]1^5 + 3 * 1 + 6 = 1 + 3 + 6 = 10.[/tex]

[tex]\therefore f(1) = 10 .[/tex]

## Bookmarks