Get comprehensive solutions to your questions with the help of IDNLearn.com's experts. Discover prompt and accurate responses from our experts, ensuring you get the information you need quickly.

Resolve into factors.

a. [tex]x^4 + x^2y^2 + y^4[/tex]


Sagot :

Certainly! Let's factor the polynomial expression [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex]:

1. Identify Polynomial Structure: Notice that the given polynomial, [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex], is symmetric in [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and can be thought of in terms of [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex].

2. Look for Patterns: By examining the polynomial, we can see it resembles the sum of cubes if we group it carefully. Let's explore possible factorizations to find the specific pattern:

3. Symmetry Observations: The expression [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex] can be viewed using the identity for the sum of cubes broken into symmetrical parts of square terms. Notice how the polynomial can be expressed as products involving polynomials of [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex].

4. Combination of Terms: Let's guess the factorization structure. We recall that certain symmetric polynomials can be factored into symmetrical parts.

Consider:
[tex]\[ (x^2 - x y + y^2)(x^2 + x y + y^2) \][/tex]

5. Verification: To ensure correctness, let's expand the presumed factorization:
[tex]\[ (x^2 - x y + y^2)(x^2 + x y + y^2) \][/tex]

We multiply the terms:
[tex]\[ = x^2(x^2 + x y + y^2) - x y(x^2 + x y + y^2) + y^2(x^2 + x y + y^2) \][/tex]
[tex]\[ = x^4 + x^3 y + x^2 y^2 - x^3 y - x^2 y^2 - x y^3 + x^2 y^2 + x y^3 + y^4 \][/tex]

6. Simplification: Combine the like terms:
[tex]\[ x^4 + (x^3 y - x^3 y) + (x^2 y^2 - x^2 y^2 + x^2 y^2) - x y^3 + x y^3 + y^4 + (x y^3 - x y^3) \][/tex]
[tex]\[ = x^4 + x^2 y^2 + y^4 \][/tex]

Thus, the given polynomial [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex] can be factored as:
[tex]\[ (x^2 - x y + y^2)(x^2 + x y + y^2) \][/tex]

Therefore, the resolved factors of the polynomial [tex]\( x^4 + x^2 y^2 + y^4 \)[/tex] are:
[tex]\[ (x^2 - x y + y^2)(x^2 + x y + y^2) \][/tex]