I'm trying to find the complex zeros of a polynomial function, x^3-8x^2+25x-26, i have found that x-2 is a factor, but when i use synthetic division, i do not end up with a remainder of zero. What does this mean? Am I doing something wrong or what?
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It means you are making a mistake in the division. x - 2 is a factor. I prefer long division. When I divide the polynomial by x - 2, I get
x^3 - 8x² + 25x - 26 = (x - 2)(x² - 6x + 13) = (x - 2)(x² - 6 + 9 + 4) = (x - 2)((x - 3)² + 4).
The other roots can be found by setting (x - 3)² + 4 = 0. These roots are complex.
x^3 - 8x² + 25x - 26 = (x - 2)(x² - 6x + 13) = (x - 2)(x² - 6 + 9 + 4) = (x - 2)((x - 3)² + 4).
The other roots can be found by setting (x - 3)² + 4 = 0. These roots are complex.
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f(2) = 8 - 8(4) + 25(2) - 26 = 0, so x - 2 is indeed a factor
2 | 1 ... -8 ..... 25 .... -26
.......... 2 ...... -12 .....26
...--------------- ----------------
....1 ...-6 ....... 13 ...... 0 <== remainder is indeed 0, so you must have made a mistake
quotient is x^2 - 6x + 13
now use the quadratic formula with a = 1 , b = -6, and c = 13 to find the other two zeros
2 | 1 ... -8 ..... 25 .... -26
.......... 2 ...... -12 .....26
...--------------- ----------------
....1 ...-6 ....... 13 ...... 0 <== remainder is indeed 0, so you must have made a mistake
quotient is x^2 - 6x + 13
now use the quadratic formula with a = 1 , b = -6, and c = 13 to find the other two zeros