Question 827552: Find the nth-degree polynomial function with real coefficients satisfying the conditions: n=3; 6 and -5+2i are zeros; f(2)=-636 Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website! A third degree polynomial with real coefficients will have three zeros. And we will need all three to find the desired polynomial.
But we've only been told what two zeros are: 6 and -5+2i. The "trick" is to know that if a polynomial with real coefficients has a complex zero, then the complex conjugate of that zero will also be a zero. (The complex conjugate of a + bi is a - bi (or a + (-b)i).) So the complex conjugate of -5+2i (which is -5-2i) will be the third zero.
A general form for a factored third degree polynomial is:
where "a" is some non-zero constant and the z's are the zeros of the polynomial. Substituting in our zeros we get:
Simplifying inside each factor we get:
We'll figure out what number to use for "a" a little later. Right now we are going to start multiplying this out. Multiplying with the complex numbers can be tricky. But here's a suggestion that should make this easier:
Multiplication is commutative. (This means the multiplications can take place in any order you choose and the result will be the same.) Take advantage of this by choosing to multiply the factors with the complex conjugates first.
Use the pattern to multiply the factors with the complex conjugates. (This pattern can always be used with these types of factors.)
So we will be smart and multiply the last two factors first. And by clever grouping within these factors:
we can see how to use the pattern to do the multiplication. Using the "(x+5)" as the "a" in the pattern and the "2i" as the "b", the last two factors will fit the (a+b)(a-b) of the pattern. And the pattern tells us how it works out ():
We can use yet another pattern, , to square the (x+5): And since by definition . Substituting this in we now have:
Simplifying...
Notice how the i's are all gone! By choosing the multiply in the order we chose, we eliminated the i's as hast as possible. If we had chosen a different order the i's would hang around, complicating things, until all the multiplications have been finished!
Next we multiply the last two factors. There is no pattern for this. We just multiply each term of one factor times each term of the other factor:
Simplifying...
Finally we will get around to figuring out the number for "a". We were given that f(2) = -636. Substituting these numbers into the equation above (the 2 for x and the -636 for f(x))) we get:
Simplifying...
Dividing by -212:
which simplifies to:
With this value for "a" our function is now:
All that is left is to distribute the 3:
This is the desired polynomial.
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