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melisa1 [442]
3 years ago
5

write a polynomial function of least degree with integral coefficients that has the given zeros. -(1/3), -i

Mathematics
1 answer:
inessss [21]3 years ago
4 0

Answer:

f(x)=3x^3+x^2+3x+1

Step-by-step explanation:

If a real number -\frac{1}{3} is a zero of polynomial function, then

x-\left(-\dfrac{1}{3}\right)=x+\dfrac{1}{3}

is the factor of this function.

If a complex number -i is a xero of the polynomial function, then the complex number i is also a zero of this function and

x-(-i)=x+i\ \text{ and }\ x-i

are two factors of this function.

So, the function of least degree is

f(x)=\left(x+\dfrac{1}{3}\right)(x+i)(x-i)=\left(x+\dfrac{1}{3}\right)(x^2-i^2)=\\ \\ =\left(x+\dfrac{1}{3}\right)(x^2+1)=\dfrac{1}{3}(3x+1)(x^2+1)=\dfrac{1}{3}(3x^3+x^2+3x+1)

If the polynomial function must be with integer coefficients, then it has a form

f(x)=3x^3+x^2+3x+1

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