Find the complex zeros of the polynomial function. Write f in factored form. f(x)=x^3-27Use the complex zeros to write f in in factored form. f(x)=? (Reduce fractions and simplify roots. Do not...

Find the complex zeros of the polynomial function. Write f in factored form.

f(x)=x^3-27

Use the complex zeros to write f in in factored form.

f(x)=?

(Reduce fractions and simplify roots. Do not enter decimals.)

Asked on by amber756

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

Factor `f(x)=x^3-27` :

We can use the difference of two cubes to factor as:

`f(x)=(x-3)(x^2+3x+9)`

To find the zeros we can set f(x)=0:

`(x-3)(x^2+3x+9)=0` Using the zero product property either (x-3)=0 or `x^2+3x+9=0` . The first gives a real zero of 3; we can use the quadratic formula to get the other two zeros:

`x^2+3x+9=0==> x=(-3+-sqrt(9-4(1)(9)))/2`

Thus `x=-3/2+-3/2isqrt(3)` are the two remaining zeros.

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The fully factored form of `f(x)=x^3-27` is:

`f(x)=(x-3)(x+3/2+3/2isqrt(3))(x+3/2-3/2isqrt(3))`

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