Answer :
Sure! Let's break down each of the given expressions into their factored forms, step by step:
a) Factor [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex]:
1. The expression is symmetric and can be viewed as a sum of squares of the form [tex]\( (x^2 + y^2)^2 - x^2 y^2 \)[/tex].
2. This can be factored using the formula for a sum of squares:
[tex]\( x^4 + x^2 y^2 + y^4 = (x^2 + y^2 - xy)(x^2 + y^2 + xy) \)[/tex].
b) Factor [tex]\( x^4 + x^2 + 1 \)[/tex]:
1. Notice that this is a trinomial and can be split into two quadratic forms by recognizing it fits a pattern.
2. It can be factored as:
[tex]\( x^4 + x^2 + 1 = (x^2 - x + 1)(x^2 + x + 1) \)[/tex].
d) Factor [tex]\( a^4 - 7a^2b^2 + b^4 \)[/tex]:
1. This expression is of the form [tex]\( (a^2 + b^2)^2 - 9a^2b^2 \)[/tex], and you can factor it using the difference of squares method.
2. It factors as:
[tex]\( a^4 - 7a^2b^2 + b^4 = (a^2 - 3ab + b^2)(a^2 + 3ab + b^2) \)[/tex].
e) Factor [tex]\( x^4 - 3x^2y^2 + 9y^4 \)[/tex]:
1. This expression is similar to a quadratic form in [tex]\( x^2 \)[/tex] and can be rearranged as [tex]\( (x^2 - 3y^2)^2 - 0 \)[/tex].
2. It factors using a pattern:
[tex]\( x^4 - 3x^2y^2 + 9y^4 = (x^2 - 3xy + 3y^2)(x^2 + 3xy + 3y^2) \)[/tex].
g) Factor [tex]\( 4x^4 + 3x^2y^2 + 9y^4 \)[/tex]:
1. This can be considered as a combination of quadratic forms in powers of [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex].
2. It can be factored into:
[tex]\( 4x^4 + 3x^2y^2 + 9y^4 = (2x^2 - 3xy + 3y^2)(2x^2 + 3xy + 3y^2) \)[/tex].
h) Factor [tex]\( 25x^4 + 4x^2y^2 + 4y^4 \)[/tex]:
1. This expression is crafted with coefficients that suggest sums and differences similar to a quadratic form.
2. It factors as:
[tex]\( 25x^4 + 4x^2y^2 + 4y^4 = (5x^2 - 4xy + 2y^2)(5x^2 + 4xy + 2y^2) \)[/tex].
These factored forms provide a clearer view of the structure of each polynomial, revealing the components within them.
a) Factor [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex]:
1. The expression is symmetric and can be viewed as a sum of squares of the form [tex]\( (x^2 + y^2)^2 - x^2 y^2 \)[/tex].
2. This can be factored using the formula for a sum of squares:
[tex]\( x^4 + x^2 y^2 + y^4 = (x^2 + y^2 - xy)(x^2 + y^2 + xy) \)[/tex].
b) Factor [tex]\( x^4 + x^2 + 1 \)[/tex]:
1. Notice that this is a trinomial and can be split into two quadratic forms by recognizing it fits a pattern.
2. It can be factored as:
[tex]\( x^4 + x^2 + 1 = (x^2 - x + 1)(x^2 + x + 1) \)[/tex].
d) Factor [tex]\( a^4 - 7a^2b^2 + b^4 \)[/tex]:
1. This expression is of the form [tex]\( (a^2 + b^2)^2 - 9a^2b^2 \)[/tex], and you can factor it using the difference of squares method.
2. It factors as:
[tex]\( a^4 - 7a^2b^2 + b^4 = (a^2 - 3ab + b^2)(a^2 + 3ab + b^2) \)[/tex].
e) Factor [tex]\( x^4 - 3x^2y^2 + 9y^4 \)[/tex]:
1. This expression is similar to a quadratic form in [tex]\( x^2 \)[/tex] and can be rearranged as [tex]\( (x^2 - 3y^2)^2 - 0 \)[/tex].
2. It factors using a pattern:
[tex]\( x^4 - 3x^2y^2 + 9y^4 = (x^2 - 3xy + 3y^2)(x^2 + 3xy + 3y^2) \)[/tex].
g) Factor [tex]\( 4x^4 + 3x^2y^2 + 9y^4 \)[/tex]:
1. This can be considered as a combination of quadratic forms in powers of [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex].
2. It can be factored into:
[tex]\( 4x^4 + 3x^2y^2 + 9y^4 = (2x^2 - 3xy + 3y^2)(2x^2 + 3xy + 3y^2) \)[/tex].
h) Factor [tex]\( 25x^4 + 4x^2y^2 + 4y^4 \)[/tex]:
1. This expression is crafted with coefficients that suggest sums and differences similar to a quadratic form.
2. It factors as:
[tex]\( 25x^4 + 4x^2y^2 + 4y^4 = (5x^2 - 4xy + 2y^2)(5x^2 + 4xy + 2y^2) \)[/tex].
These factored forms provide a clearer view of the structure of each polynomial, revealing the components within them.
