SOLUTION: Find a polynomial f(x) of degree 3 with real coefficients and the following zeros. -2, 2

SOLUTION: Find a polynomial f(x) of degree 3 with real coefficients and the following zeros. -2, 2 - i

Question 824327: Find a polynomial f(x) of degree 3 with real coefficients and the following zeros.
-2, 2 - i

Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
A polynomial of degree 3 should have 3 zeros. But only two have been given. The third zero must be found in order to find the polynomial.

The key to finding the third zero is: If a polynomial with real coefficients has complex zeros, then they will always come in conjugate pairs. Since we were given one complex zero, 2 - i, then the missing zero must be its conjugate: 2 + i.

So the three zeros are -2, 2 - i and 2 + i. And when a number, let's call it "z", is a zero of a polynomial, then (x - z) is a factor of the polynomial. So, in factored form,

Simplifying each factor we get:

All that is left is to multiply this out. (Hint for future problems: Multiply the factors with complex conjugate zeros together first.) Multiplying the last two factors can be done with a clever use of the . To show you how I will do some grouping within those factors:

Treating the "(x-2)" as the "a" of the pattern and the "i" as the "b", this pattern tells us that multiplying the last two factors will result in :

We can use another pattern, to square the (x-2). And . So we get:

which simplifies as follows:

Last we multiply the remaining factors (which I will leave to you). Just multiply each term of (x+2) times each term of . (That's 6 multiplications!) Then add any like terms. Then put the terms in standard (highest exponent to lowest exponent) order.
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