|
Question 73367: FIND A THIRD-DEGREE POLYNOMIAL EQUATION WITH RATIONAL COEFFICIENTS THAT HAS ROOTS -5 AND 6+i
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Complex roots occur only in pairs, so the roots are actually 6 - i and 6 + 1 and -5.
.
Putting these into factor form you get:
.
(x + 5)*(x - (6 + i))*(x - (6 - i))
.
All you have to do now is multiply all of these together and you will have the answer.
.
Let's first work on multiplying the two complex terms. First remove the parentheses
by changing the signs of the terms within. When you do the two factors that you will be
multiplying are:
.
(x - 6 - i)* (x - 6 + i)
.
The way to do this is to take the terms in the first set of parentheses one at a time and
multiply them by the terms in the second set of parentheses.
.
So select the x in the first set of parentheses and multiply it by the x, then the –6, and then
the +i from the second set of parentheses to get three answer terms consisting of
.
x^2–6x+xi
.
Then select the –6 in the first set of parentheses and multiply it by the x, then the –6, and
then the +i from the second set of parentheses to get three more answer terms consisting of
.
-6x + 36 –6i
.
Finally, take the –i from the first set of parentheses and multiply it times the x, the –6,
and the i in the second set of parentheses to get three more answer terms consisting of
.
-xi + 6i –i^2
.
One more thing. Recall that by definition i^2 = -1 and
if you
substitute –1 for i^2 in the three answer terms of this group, you get
-xi + 6i –(-1)
which simplifies to -xi + 6i +1
.
Now add the three groups of answer terms, noting that some of them are equal but with
opposite signs so they cancel out. The string of answer terms before cancellation is:
.
x^2 – 6x + xi –6x + 36 + 6i – xi + 6i + 1
.
Cancel the 6i terms and the xi terms and you are left with:
.
x^2 – 6x – 6x + 36 + 1
.
When you combine like terms you have:
.
x^2 – 12x + 37
.
That takes care of multiplying two of the factors together. Now all you have to do is
multiply that trinomial by the third factor of the binomial (x + 5)
.
Multiply each term in the trinomial using the x from the binomial to get:
.
x^3 – 12x^2 + 37x
.
Then multiply each of the terms in the trinomial by the +5 from the binomial to get:
.
5x^2 – 60x + 185
.
Add these 6 multiplied terms and you get:
.
x^3 - 12x^2 + 37x + 5x^2 – 60x + 185
.
Combine the terms having like powers of x and you end up with:
.
x^3 - 7x^2 – 23x + 185
.
This is the third degree polynomial you were looking for. In equation form it would be:
.
x^3 - 7x^2 – 23x + 185 = 0
.
Hope this is helpful to you and that you can follow your way down this long and winding path.
|
|
|
| |
