Timmy solved the polynomial equation x^5 -x^4-2x^3+2x^2-24x+24=0 and his solutions were x=-1,√6, 2i. Explain his error(s) and list the correct solutions.

Answer 1

The Fundamental Theorem of Algebra says that a 5th order polynomial will have 5 solutions so, since only 3 are shown we know there is a problem.

First, let's verify that -1 is a solution by plugging it in:

(-1)⁵ - (-1)⁴ - 2(-1)³ + 2(-1)² - 24(-1) + 24 = -1 -1 - -2 + 2 - -24 + 24 = 50 so that is a mistake.

Let's try √6:

(√6)⁵ - (√6)⁴ - 2(√6)³ + 2(√6)² - 24(√6) + 24 = 36√6 - 36 - 12√6 + 12 - 24√6 + 24 = 0 so this truly is a solution. Let's try the conjugate of √6 which is -√6:

(-√6)⁵ - (-√6)⁴ - 2(-√6)³ + 2(-√6)² - 24(-√6) + 24 = -36√6 - 36 + 12√6 + 12 + 24√6 + 24 = 0 so this is also a solution.

Let's try i (or √-1

(√-1)⁵ - (√-1)⁴ - 2(√-1)³ + 2(√-1)² - 24(√-1) + 24 = √-1 - 1 + 2√-1 - 2 - 24√-1 + 24 = 21 - 21√-1 so this is a mistake as well

So far, we know that √6 and -√6 are solutions. From here it's a matter of guesswork to try possible rational solutions using the rational roots theorem or other tricks like that but in today's world, the easier thing is to graph it and look to see if we can learn anything about the solutions. The graph looks like this:

I can see the +/-√6 and it also looks like 1 might be a solution. So let's try 1:

(1)⁵ - (1)⁴ - 2(1)³ + 2(1)² - 24(1) + 24 = 1 -1 - 2 + 2 - 24 + 24 = 0 so it is a solution.

The other two solutions are imaginary because the graph only crosses the x-axis at 3 spots meaning there are only 3 real solutions and, since there must be a total of 5 the other 2 must be imaginary.

Since we know that +/-√6 and 1 are solutions, we know that (x - √6) and (x + √6) and (x - 1) are factors. Multiplying these together we get x³ - x² - 6x + 6 so we can divide that into the original polynomial to get the other factor. Dividing it using long division, we get x² + 4.

Now we can set that equal to 0 and solve to get 2i and -2i as the other two solutions.