Answer :
- Rewrite each expression by adding and subtracting terms to complete the square.
- Express each completed square as a difference of squares.
- Factor each difference of squares into two binomials.
- The factored forms are: a) $(x^2+xy+y^2)(x^2-xy+y^2)$, b) $(x^2+x+1)(x^2-x+1)$, d) $(a^2+3ab+b^2)(a^2-3ab+b^2)$, e) $(x^2+3xy+3y^2)(x^2-3xy+3y^2)$, g) $(2x^2+3xy+3y^2)(2x^2-3xy+3y^2)$, h) $(5x^2+4xy+2y^2)(5x^2-4xy+2y^2)$.
### Explanation
1. Problem Analysis
We are given six expressions to factorize. Our strategy will be to complete the square by adding and subtracting terms. This will allow us to express each expression as a difference of squares, which can then be factored.
2. Factoring a)
a) $x^4+x^2 y^2+y^4$
We can rewrite this as:
$x^4+2x^2 y^2+y^4 - x^2y^2 = (x^2+y^2)^2 - (xy)^2$
Now we have a difference of squares, so we can factor it as:
$(x^2+y^2+xy)(x^2+y^2-xy)$
3. Factoring b)
b) $x^4+x^2+1$
We can rewrite this as:
$x^4+2x^2+1 - x^2 = (x^2+1)^2 - x^2$
Now we have a difference of squares, so we can factor it as:
$(x^2+1+x)(x^2+1-x)$
4. Factoring d)
d) $a^4-7 a^2 b^2+b^4$
We can rewrite this as:
$a^4+2a^2b^2 + b^4 - 9a^2b^2 = (a^2+b^2)^2 - (3ab)^2$
Now we have a difference of squares, so we can factor it as:
$(a^2+b^2+3ab)(a^2+b^2-3ab)$
5. Factoring e)
e) $x^4-3 x^2 y^2+9 y^4$
We can rewrite this as:
$x^4+6x^2y^2 + 9y^4 - 9x^2y^2 = (x^2+3y^2)^2 - (3xy)^2$
Now we have a difference of squares, so we can factor it as:
$(x^2+3y^2+3xy)(x^2+3y^2-3xy)$
6. Factoring g)
g) $4 x^4+3 x^2 y^2+9 y^4$
We can rewrite this as:
$4x^4 + 12x^2y^2 + 9y^4 - 9x^2y^2 = (2x^2+3y^2)^2 - (3xy)^2$
Now we have a difference of squares, so we can factor it as:
$(2x^2+3y^2+3xy)(2x^2+3y^2-3xy)$
7. Factoring h)
h) $25 x^4+4 x^2 y^2+4 y^4$
We can rewrite this as:
$25x^4 + 20x^2y^2 + 4y^4 - 16x^2y^2 = (5x^2+2y^2)^2 - (4xy)^2$
Now we have a difference of squares, so we can factor it as:
$(5x^2+2y^2+4xy)(5x^2+2y^2-4xy)$
8. Final Answer
In summary, we have factored each of the given expressions by completing the square and using the difference of squares factorization.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used extensively in engineering, physics, and computer science. For example, when designing a bridge, engineers use polynomial equations to model the forces acting on the structure. Factoring these polynomials helps them find critical points and ensure the bridge's stability. Similarly, in physics, factoring can simplify equations describing the motion of objects, making them easier to analyze. In computer graphics, factoring is used to optimize rendering algorithms, improving the efficiency of image processing.
- Express each completed square as a difference of squares.
- Factor each difference of squares into two binomials.
- The factored forms are: a) $(x^2+xy+y^2)(x^2-xy+y^2)$, b) $(x^2+x+1)(x^2-x+1)$, d) $(a^2+3ab+b^2)(a^2-3ab+b^2)$, e) $(x^2+3xy+3y^2)(x^2-3xy+3y^2)$, g) $(2x^2+3xy+3y^2)(2x^2-3xy+3y^2)$, h) $(5x^2+4xy+2y^2)(5x^2-4xy+2y^2)$.
### Explanation
1. Problem Analysis
We are given six expressions to factorize. Our strategy will be to complete the square by adding and subtracting terms. This will allow us to express each expression as a difference of squares, which can then be factored.
2. Factoring a)
a) $x^4+x^2 y^2+y^4$
We can rewrite this as:
$x^4+2x^2 y^2+y^4 - x^2y^2 = (x^2+y^2)^2 - (xy)^2$
Now we have a difference of squares, so we can factor it as:
$(x^2+y^2+xy)(x^2+y^2-xy)$
3. Factoring b)
b) $x^4+x^2+1$
We can rewrite this as:
$x^4+2x^2+1 - x^2 = (x^2+1)^2 - x^2$
Now we have a difference of squares, so we can factor it as:
$(x^2+1+x)(x^2+1-x)$
4. Factoring d)
d) $a^4-7 a^2 b^2+b^4$
We can rewrite this as:
$a^4+2a^2b^2 + b^4 - 9a^2b^2 = (a^2+b^2)^2 - (3ab)^2$
Now we have a difference of squares, so we can factor it as:
$(a^2+b^2+3ab)(a^2+b^2-3ab)$
5. Factoring e)
e) $x^4-3 x^2 y^2+9 y^4$
We can rewrite this as:
$x^4+6x^2y^2 + 9y^4 - 9x^2y^2 = (x^2+3y^2)^2 - (3xy)^2$
Now we have a difference of squares, so we can factor it as:
$(x^2+3y^2+3xy)(x^2+3y^2-3xy)$
6. Factoring g)
g) $4 x^4+3 x^2 y^2+9 y^4$
We can rewrite this as:
$4x^4 + 12x^2y^2 + 9y^4 - 9x^2y^2 = (2x^2+3y^2)^2 - (3xy)^2$
Now we have a difference of squares, so we can factor it as:
$(2x^2+3y^2+3xy)(2x^2+3y^2-3xy)$
7. Factoring h)
h) $25 x^4+4 x^2 y^2+4 y^4$
We can rewrite this as:
$25x^4 + 20x^2y^2 + 4y^4 - 16x^2y^2 = (5x^2+2y^2)^2 - (4xy)^2$
Now we have a difference of squares, so we can factor it as:
$(5x^2+2y^2+4xy)(5x^2+2y^2-4xy)$
8. Final Answer
In summary, we have factored each of the given expressions by completing the square and using the difference of squares factorization.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used extensively in engineering, physics, and computer science. For example, when designing a bridge, engineers use polynomial equations to model the forces acting on the structure. Factoring these polynomials helps them find critical points and ensure the bridge's stability. Similarly, in physics, factoring can simplify equations describing the motion of objects, making them easier to analyze. In computer graphics, factoring is used to optimize rendering algorithms, improving the efficiency of image processing.
