2X^4-9X^3+3X^2-X-5=0
Given that one root is 2+i, solve the equation.
So far I have it set up like this:
2X^4-9X^3+3X^2-X-5/X^2-4X+5
I don't know how to get the answer, I know that I have to use long division but when I started to try it I kept getting weird answers. Please Help
I think there is something wrong here,
either you have a typo in the opening equation, or the question is faulty.
If one root is 2+i, then 2-i must be another root
You correctly determined that
x^2 - 4x + 5 must then be a factor
When I divided that into the original, I got an answer of 2x^2 - x -11 with a remainder of -40x + 50
my bad the original equation is:
2X^4 - 9X^3 + 13X^2 - X - 5 = 0
Ok, this time my division was exact, as expected,
and I got 2x^2 - x - 1, which factors to
(2x + 1)(x - 1)
so the roots are
2+i, 2-i, -1/2, and 1
oh I get it thanks
To solve the equation when a root is given, you can use the fact that complex roots always come in conjugate pairs. In this case, if 2+i is a root, then its conjugate, 2-i, must also be a root.
Now, we can use synthetic division to find the quotient when dividing the given polynomial by (X - (2+i))(X - (2-i)).
The synthetic division process is as follows:
1. Set up a synthetic division table with the coefficients of the polynomial:
2 -9 3 -1 -5
2. Using the complex root 2+i, write it as (X - (2+i)):
(X - (2+i))
3. Write the conjugate of the complex root, which is 2-i:
(X - (2-i))
4. Start the synthetic division process:
Write the first coefficient (2) in the top row of the table.
5. Multiply the complex root with the first coefficient (2+i) * 2:
(2+i)*2 = 4+2i
6. Write the result (4+2i) in the second row of the table.
7. Add the second row coefficient to the second coefficient of the polynomial (-9 + 4+2i = -5+2i):
-5+2i
8. Multiply this result by the complex root (2+i):
(-5+2i)*(2+i) = (-5*2 + 2i*2 + 2i*(-5) + 2i*i) = -10 + 4i - 10i - 2 = -12 - 6i
9. Write the result (-12 - 6i) in the third row of the table.
10. Add the third row coefficient to the third coefficient of the polynomial (3 + -12 - 6i = -9 - 6i):
-9 - 6i
11. Multiply this result by the complex root (2+i):
(-9 - 6i)*(2+i) = (-9*2 + -6i*2 + -6i*(-9) + -6i*i) = -18 - 12i + 54i + 6 = -12 + 42i - 6 = -18 + 42i
12. Write the result (-18 + 42i) in the fourth row of the table.
13. Add the fourth row coefficient to the fourth coefficient of the polynomial (-1 + -18 + 42i = -19 + 42i):
-19 + 42i
14. Multiply this result by the complex root (2+i):
(-19 + 42i)*(2+i) = (-19*2 + 42i*2 + 42i*(-19) + 42i*i) = -38 + 84i + -798i + -42 = -80 - 714i
15. Write the result (-80 - 714i) in the fifth row of the table.
16. Add the fifth row coefficient to the fifth coefficient of the polynomial (-5 + -80 - 714i = -85 - 714i):
-85 - 714i
Now, we have completed synthetic division.
The resulting table looks like this:
2 | -9 3 -1 -5
| 4+2i -12-6i -18+42i -80-714i
| -9-6i -19+42i -85-714i
From the synthetic division table, we can see that the quotient is given by:
-9 - 6i -19 + 42i - 85 - 714i
Thus, the equation factored with one root as 2+i is:
(X - (2+i))(X - (2-i))(X - (-9 - 6i))(X - (-19 + 42i))(X - (-85 - 714i)) = 0
You can further simplify the equation and solve for the other roots by setting each factor of the equation equal to zero.