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How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 8, -i, i?

Answer 1

Caleb Alexander

#x^3-8x^2+x-6#

The polynomial with zeros #x_1, x_2 and x_3# is

#x^3-(x_1+x_2+x_3)x^2+(x_2x_3+x_3x_1+x_1x_2)x-x_1x_2x_3#

Here, it is

#x^3-(8+i-i)x^2+(-i^2-8i+8i)x-(9)(i)(-i)#

#=x^3-8x^2+x-6#

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Answer 2

Lily Anderson

To write a polynomial function of least degree and leading coefficient 1 when the zeros are (8), (-i), and (i), we use the fact that complex zeros come in conjugate pairs for polynomials with real coefficients.

The polynomial function can be expressed as:

[f(x) = (x - 8)(x + i)(x - i)]

Expanding this expression:

[f(x) = (x - 8)(x^2 + ix - ix - i^2)]

[f(x) = (x - 8)(x^2 - i^2)]

[f(x) = (x - 8)(x^2 + 1)]

Now, multiply the factors:

[f(x) = x(x^2 + 1) - 8(x^2 + 1)]

[f(x) = x^3 + x - 8x^2 - 8]

Combine like terms:

[f(x) = x^3 - 8x^2 + x - 8]

So, the polynomial function of least degree and leading coefficient 1 with zeros (8), (-i), and (i) is (f(x) = x^3 - 8x^2 + x - 8).

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Answer from HIX Tutor

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.