Answer :
To find a polynomial in standard form with the given zeros, you need to follow these steps:
Given Zeros:
- [tex]\( -3i \)[/tex]
- [tex]\( \sqrt{3} \)[/tex]
When you have complex or irrational zeros, their conjugates are also zeros of the polynomial. Therefore, the zeros are:
- [tex]\( -3i \)[/tex]
- [tex]\( 3i \)[/tex] (conjugate of [tex]\( -3i \)[/tex])
- [tex]\( \sqrt{3} \)[/tex]
- [tex]\( -\sqrt{3} \)[/tex] (conjugate of [tex]\( \sqrt{3} \)[/tex])
Form the polynomial from the zeros:
1. Write each zero as a factor in the form [tex]\( (x - \text{zero}) \)[/tex]:
- [tex]\( (x + 3i) \)[/tex]
- [tex]\( (x - 3i) \)[/tex]
- [tex]\( (x - \sqrt{3}) \)[/tex]
- [tex]\( (x + \sqrt{3}) \)[/tex]
2. Pair up and multiply the conjugate pairs to simplify:
- Multiply [tex]\( (x + 3i)(x - 3i) \)[/tex]:
[tex]\[
(x + 3i)(x - 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9
\][/tex]
- Multiply [tex]\( (x - \sqrt{3})(x + \sqrt{3}) \)[/tex]:
[tex]\[
(x - \sqrt{3})(x + \sqrt{3}) = x^2 - (\sqrt{3})^2 = x^2 - 3
\][/tex]
3. Combine the results:
- Multiply the two quadratics obtained:
[tex]\[
(x^2 + 9)(x^2 - 3)
\][/tex]
- First distribute across the terms:
[tex]\[
= x^2 \cdot x^2 + x^2 \cdot (-3) + 9 \cdot x^2 + 9 \cdot (-3)
\][/tex]
- Simplify:
[tex]\[
= x^4 - 3x^2 + 9x^2 - 27
\][/tex]
- Combine like terms:
[tex]\[
= x^4 + 6x^2 - 27
\][/tex]
Therefore, the polynomial in standard form is [tex]\( f(x) = x^4 + 6x^2 - 27 \)[/tex].
The correct answer is A) [tex]\( f(x) = x^4 + 6x^2 - 27 \)[/tex].
Given Zeros:
- [tex]\( -3i \)[/tex]
- [tex]\( \sqrt{3} \)[/tex]
When you have complex or irrational zeros, their conjugates are also zeros of the polynomial. Therefore, the zeros are:
- [tex]\( -3i \)[/tex]
- [tex]\( 3i \)[/tex] (conjugate of [tex]\( -3i \)[/tex])
- [tex]\( \sqrt{3} \)[/tex]
- [tex]\( -\sqrt{3} \)[/tex] (conjugate of [tex]\( \sqrt{3} \)[/tex])
Form the polynomial from the zeros:
1. Write each zero as a factor in the form [tex]\( (x - \text{zero}) \)[/tex]:
- [tex]\( (x + 3i) \)[/tex]
- [tex]\( (x - 3i) \)[/tex]
- [tex]\( (x - \sqrt{3}) \)[/tex]
- [tex]\( (x + \sqrt{3}) \)[/tex]
2. Pair up and multiply the conjugate pairs to simplify:
- Multiply [tex]\( (x + 3i)(x - 3i) \)[/tex]:
[tex]\[
(x + 3i)(x - 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9
\][/tex]
- Multiply [tex]\( (x - \sqrt{3})(x + \sqrt{3}) \)[/tex]:
[tex]\[
(x - \sqrt{3})(x + \sqrt{3}) = x^2 - (\sqrt{3})^2 = x^2 - 3
\][/tex]
3. Combine the results:
- Multiply the two quadratics obtained:
[tex]\[
(x^2 + 9)(x^2 - 3)
\][/tex]
- First distribute across the terms:
[tex]\[
= x^2 \cdot x^2 + x^2 \cdot (-3) + 9 \cdot x^2 + 9 \cdot (-3)
\][/tex]
- Simplify:
[tex]\[
= x^4 - 3x^2 + 9x^2 - 27
\][/tex]
- Combine like terms:
[tex]\[
= x^4 + 6x^2 - 27
\][/tex]
Therefore, the polynomial in standard form is [tex]\( f(x) = x^4 + 6x^2 - 27 \)[/tex].
The correct answer is A) [tex]\( f(x) = x^4 + 6x^2 - 27 \)[/tex].
