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Question 1208646: Form a polynomial f left parenthesis x right parenthesis with real coefficients having the given degree and zeros.
Degree 5; zeros: negative 4 ; minus i ; 5 plus i
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer is f(x) = x^5 - 6x^4 - 13x^3 + 98x^2 - 14x + 104
How to find that answer:
Since all of the coefficients are real numbers, this means the complex roots come in conjugate pairs.
a+bi pairs with a-bi
-i pairs with i
5+i pairs with 5-i
The five roots are: -4, i, -i, 5+i, 5-i
Recall the fundamental theorem of algebra says that any nth degree polynomial has n complex roots.
x = i has both sides square to x^2 = -1 and then we get everything to one side: x^2+1=0. So that yields the factor (x^2+1).
x = 5+i becomes x-5 = i; then both sides square to x^2-10x+25 = -1 and it becomes x^2-10x+26=0
Use the quadratic formula to solve x^2-10x+26=0 and you should get x = 5+i and x = 5-i.
Note: If you're using GeoGebra, you need to use the CSolve command (in contrast to the regular Solve command). Otherwise, it will produce an empty set of solutions.
That slight tangent aside, we can say the following
The root x = -4 leads to the factor (x+4)
The roots x = i, x = -i lead to the factor (x^2+1)
The roots x = 5+i, x = 5-i lead to the factor (x^2-10x+26)
The goal is to expand this out: (x+4)(x^2+1)(x^2-10x+26)
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For now let's focus on expanding the portion (x+4)(x^2+1)
We could use the FOIL rule, but I'll use the box method instead.
Place the terms x and 4 along the left hand side. Place the terms x^2 and 1 along the top.
To fill out this table, multiply each pair of headers.
Eg: top left corner is x^2*x = x^3
The inner terms in blue are then added to get x^3+4x^2+x+4. There aren't any like terms to combine.
We have shown that (x+4)(x^2+1) = x^3+4x^2+x+4
So
(x+4)(x^2+1)(x^2-10x+26)
updates to
(x^3+4x^2+x+4)(x^2-10x+26)
We'll need to do one more application of the box method.
Here's the blank template with the headers filled in.
And here's the completed table
| x^2 | -10x | 26 | | x^3 | x^5 | -10x^4 | 26x^3 | | 4x^2 | 4x^4 | -40x^3 | 104x^2 | | x | x^3 | -10x^2 | 26x | | 4 | 4x^2 | -40x | 104 |
Add up the terms in blue.
This time we have groups of like terms to combine (eg: 4x^4 + (-10x^4) = -6x^4)
Notice the like terms are along northeast diagonals.
I'll skip a bit of scratch work and leave it to the student.
You should get the final result f(x) = x^5 - 6x^4 - 13x^3 + 98x^2 - 14x + 104
You can use various software tools to verify this answer.
WolframAlpha is one example. GeoGebra is another (make sure to use Csolve instead of Solve).
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