SOLUTION: Find a polynomial function of least degree having only real​ coefficients, a leading coefficient of​ 1, and zeros of 3 and 2 plus i .

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Question 1126012: Find a polynomial function of least degree having only real​ coefficients, a leading coefficient of​ 1, and zeros of 3 and 2 plus i .
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

if zeros are (real one) and (complex one), then you also have because complex zeros always come in pairs
since you have three zeros, a polynomial function of least degree will be a polynomial function of degree
...substitute given zeros




Answer by greenestamps(13215)   (Show Source): You can put this solution on YOUR website!


Given the three roots p, q, and r, one way you can find the equation is multiply out (x-p)(x-q)(x-r), as shown by the other tutor. With the two complex roots, the algebra gets a bit ugly.

So here is another way to find the equation that some students might find easier.

In the final equation (with leading coefficient 1)

,

the coefficient of the quadratic term is the opposite of the sum of the roots:


the coefficient of the linear term is the sum of the products of the roots two at a time:


and the constant term is the opposite of the product of the three roots:


With roots 3, 2+i, and 2-i, we have...

so b = -7;

so c = 17; and

so d = -15.

Then the polynomial is
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