# How do you write a polynomial in standard form given the zeros x=1, -1, √3i, -√3i?

The polynomial will be:

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To write a polynomial in standard form given the zeros ( x = 1, -1, \sqrt{3}i, -\sqrt{3}i ), you can use the fact that complex roots occur in conjugate pairs.

Start by writing the factors of the polynomial using the given zeros. For real roots ( x = 1 ) and ( x = -1 ), the factors are ( (x - 1) ) and ( (x + 1) ), respectively.

For the complex conjugate pair ( x = \sqrt{3}i ) and ( x = -\sqrt{3}i ), the factors are ( (x - \sqrt{3}i) ) and ( (x + \sqrt{3}i) ), respectively.

Now, multiply all these factors together to obtain the polynomial expression.

[ (x - 1)(x + 1)(x - \sqrt{3}i)(x + \sqrt{3}i) ]

Expand this expression and simplify to get the polynomial in standard form.

[ (x - 1)(x + 1)(x^2 + 3) ]

Now, distribute and multiply the factors:

[ (x - 1)(x + 1)(x^2 + 3) = (x^2 - 1)(x^2 + 3) ]

[ = x^4 + 3x^2 - x^2 - 3 ]

[ = x^4 + 2x^2 - 3 ]

Thus, the polynomial in standard form is ( x^4 + 2x^2 - 3 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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