Answer :
To determine which expression could represent a polynomial with a factor of [tex]\( (x-\sqrt{3}i) \)[/tex], we follow specific steps to solve the question systematically.
First, understand that if [tex]\( (x-\sqrt{3}i) \)[/tex] is a factor of a polynomial, then its complex conjugate, [tex]\( (x+\sqrt{3}i) \)[/tex], must also be a factor. Therefore, any polynomial with [tex]\( (x-\sqrt{3}i) \)[/tex] as a factor must also include [tex]\( (x+\sqrt{3}i) \)[/tex] as a factor. This results in a quadratic factor of the form:
[tex]\[ (x-\sqrt{3}i)(x+\sqrt{3}i) = x^2 - (\sqrt{3}i)^2 = x^2 + 3 \][/tex]
Given this quadratic factor, we now analyze the given polynomial options to see which ones can be expressed as a product involving [tex]\( (x^2 + 3) \)[/tex]:
1. [tex]\( 4x^4 + 11x^2 - 3 \)[/tex]
2. [tex]\( 3x^4 + 26x^2 - 9 \)[/tex]
3. [tex]\( 4x^4 - 11x^2 + 3 \)[/tex]
4. [tex]\( 3x^4 - 26x^2 - 9 \)[/tex]
We'll substitute each polynomial and see if [tex]\( x^2 + 3 \)[/tex] can be a factor.
### Analysis of each polynomial:
1. For [tex]\( 4x^4 + 11x^2 - 3 \)[/tex]:
- Try to factorize and see if [tex]\( x^2 + 3 \)[/tex] is a factor:
It does not simplify to include [tex]\( x^2 + 3 \)[/tex].
2. For [tex]\( 3x^4 + 26x^2 - 9 \)[/tex]:
- Try to factorize and see if [tex]\( x^2 + 3 \)[/tex] is a factor:
It does not simplify to include [tex]\( x^2 + 3 \)[/tex].
3. For [tex]\( 4x^4 - 11x^2 + 3 \)[/tex]:
- Try to factorize and see if [tex]\( x^2 + 3 \)[/tex] is a factor:
It does not simplify to include [tex]\( x^2 + 3 \)[/tex].
4. For [tex]\( 3x^4 - 26x^2 - 9 \)[/tex]:
- Try to factorize and see if [tex]\( x^2 + 3 \)[/tex] is a factor:
It does not simplify to include [tex]\( x^2 + 3 \)[/tex].
None of the given polynomials can be factored to include [tex]\( x^2 + 3 \)[/tex].
Therefore, the expression with a factor of [tex]\( (x-\sqrt{3}i) \)[/tex] is not among the given polynomials.
First, understand that if [tex]\( (x-\sqrt{3}i) \)[/tex] is a factor of a polynomial, then its complex conjugate, [tex]\( (x+\sqrt{3}i) \)[/tex], must also be a factor. Therefore, any polynomial with [tex]\( (x-\sqrt{3}i) \)[/tex] as a factor must also include [tex]\( (x+\sqrt{3}i) \)[/tex] as a factor. This results in a quadratic factor of the form:
[tex]\[ (x-\sqrt{3}i)(x+\sqrt{3}i) = x^2 - (\sqrt{3}i)^2 = x^2 + 3 \][/tex]
Given this quadratic factor, we now analyze the given polynomial options to see which ones can be expressed as a product involving [tex]\( (x^2 + 3) \)[/tex]:
1. [tex]\( 4x^4 + 11x^2 - 3 \)[/tex]
2. [tex]\( 3x^4 + 26x^2 - 9 \)[/tex]
3. [tex]\( 4x^4 - 11x^2 + 3 \)[/tex]
4. [tex]\( 3x^4 - 26x^2 - 9 \)[/tex]
We'll substitute each polynomial and see if [tex]\( x^2 + 3 \)[/tex] can be a factor.
### Analysis of each polynomial:
1. For [tex]\( 4x^4 + 11x^2 - 3 \)[/tex]:
- Try to factorize and see if [tex]\( x^2 + 3 \)[/tex] is a factor:
It does not simplify to include [tex]\( x^2 + 3 \)[/tex].
2. For [tex]\( 3x^4 + 26x^2 - 9 \)[/tex]:
- Try to factorize and see if [tex]\( x^2 + 3 \)[/tex] is a factor:
It does not simplify to include [tex]\( x^2 + 3 \)[/tex].
3. For [tex]\( 4x^4 - 11x^2 + 3 \)[/tex]:
- Try to factorize and see if [tex]\( x^2 + 3 \)[/tex] is a factor:
It does not simplify to include [tex]\( x^2 + 3 \)[/tex].
4. For [tex]\( 3x^4 - 26x^2 - 9 \)[/tex]:
- Try to factorize and see if [tex]\( x^2 + 3 \)[/tex] is a factor:
It does not simplify to include [tex]\( x^2 + 3 \)[/tex].
None of the given polynomials can be factored to include [tex]\( x^2 + 3 \)[/tex].
Therefore, the expression with a factor of [tex]\( (x-\sqrt{3}i) \)[/tex] is not among the given polynomials.
