Answer :
Sure! Let's resolve each of these expressions into their factors step-by-step:
12. Factor [tex]\( 4x^4 + 81 \)[/tex]:
This is a sum of squares, which typically does not factor into simple real factors. However, it can be expressed as a product of two factors over complex numbers. In our case, the expression can be factored into:
[tex]\[
(2x^2 - 6x + 9)(2x^2 + 6x + 9)
\][/tex]
This form uses complex roots due to the sum of squares structure.
13. Factor [tex]\( 9x^4 - 3x^2 + 1 \)[/tex]:
This is a quadratic in terms of [tex]\( x^2 \)[/tex]. Here, you consider [tex]\( x^2 \)[/tex] as a single variable and factor the expression:
[tex]\[
(3x^2 - 3x + 1)(3x^2 + 3x + 1)
\][/tex]
Each factor here forms a quadratic trinomial.
14. Factor [tex]\( x^4 + x^2 + 1 \)[/tex]:
This expression is a particular form of polynomial that factors as follows:
[tex]\[
(x^2 - x + 1)(x^2 + x + 1)
\][/tex]
Notice that each factor is a quadratic and contains the variable [tex]\( x \)[/tex].
15. Factor [tex]\( x^4 + 3x^2y^2 + 4y^4 \)[/tex]:
This expression appears to be a form of a quadratic in [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex]. It can be factored as:
[tex]\[
(x^2 - xy + 2y^2)(x^2 + xy + 2y^2)
\][/tex]
Each factor highlights the mixed terms and maintains the symmetry in the powers of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
These factored forms show how the given polynomials can be broken into products of simpler expressions.
12. Factor [tex]\( 4x^4 + 81 \)[/tex]:
This is a sum of squares, which typically does not factor into simple real factors. However, it can be expressed as a product of two factors over complex numbers. In our case, the expression can be factored into:
[tex]\[
(2x^2 - 6x + 9)(2x^2 + 6x + 9)
\][/tex]
This form uses complex roots due to the sum of squares structure.
13. Factor [tex]\( 9x^4 - 3x^2 + 1 \)[/tex]:
This is a quadratic in terms of [tex]\( x^2 \)[/tex]. Here, you consider [tex]\( x^2 \)[/tex] as a single variable and factor the expression:
[tex]\[
(3x^2 - 3x + 1)(3x^2 + 3x + 1)
\][/tex]
Each factor here forms a quadratic trinomial.
14. Factor [tex]\( x^4 + x^2 + 1 \)[/tex]:
This expression is a particular form of polynomial that factors as follows:
[tex]\[
(x^2 - x + 1)(x^2 + x + 1)
\][/tex]
Notice that each factor is a quadratic and contains the variable [tex]\( x \)[/tex].
15. Factor [tex]\( x^4 + 3x^2y^2 + 4y^4 \)[/tex]:
This expression appears to be a form of a quadratic in [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex]. It can be factored as:
[tex]\[
(x^2 - xy + 2y^2)(x^2 + xy + 2y^2)
\][/tex]
Each factor highlights the mixed terms and maintains the symmetry in the powers of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
These factored forms show how the given polynomials can be broken into products of simpler expressions.
