Question

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Factorize:
1). (a⁴-8a²b²+16b⁴) - 256
2). a⁴-6a²b²+9b⁴-81
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Answer 1

1.) a²(a² - 8b²) + 16(b² + 4)(b + 2)(b -2)

2.) a²(a² - 6b²) + 9(b² + 3)(b² - 3)

Explanation:

1). (a⁴-8a²b²+16b⁴) - 256

a⁴-8a²b²+16b⁴ - 256

a²(a² - 8b²) + 16(b⁴ - 16)

a²(a² - 8b²) + 16(b² + 4)(b² - 4)

a²(a² - 8b²) + 16(b² + 4)(b + 2)(b -2)

2). a⁴-6a²b²+9b⁴-81

a²(a² - 6b²) + 9(b⁴ - 9)

a²(a² - 6b²) + 9(b² + 3)(b² - 3)

Answer 2

(1) (a² - 4b² + 16)(a² - 4b² - 16)

(2) (a² - 3b² + 9)(a² - 3b² - 9)

Explanation:

To factorize the given expressions, we'll look for patterns that can be expressed as the difference of squares and perfect squares.

`\hrulefill`

Question #1: (a⁴ - 8a²b² + 16b⁴) - 256

To factorize this expression, let's first rewrite it as follows:

⇒ (a⁴ - 8a²b² + 16b⁴) - 256 = a⁴ - 8a²b² + 16b⁴ - 256

Now, we can see that the expression inside the parentheses is a difference of squares:

⇒ a⁴ - 8a²b² + 16b⁴ = (a² - 4b²)²

And we can factorize 256 as a perfect square:

⇒ 256 = 16²

Now, let's rewrite the original expression with the factored forms:

⇒ a⁴ - 8a²b² + 16b⁴ - 256 = (a² - 4b²)² - 16²

Now, this expression can be further factorized as a difference of squares:

⇒ (a² - 4b²)² - 16² = (a² - 4b² + 16)(a² - 4b² - 16)

`\hrulefill`

Question #2: a⁴ - 6a²b² + 9b⁴ - 81

To factorize this expression, we can see that the expression inside the parentheses is a difference of squares:

⇒ a⁴ - 6a²b² + 9b⁴ = (a² - 3b²)²

And we can factorize 81 as a perfect square:

⇒ 81 = 9²

Now, let's rewrite the original expression with the factored forms:

⇒ a⁴ - 6a²b² + 9b⁴ - 81 = (a² - 3b²)² - 9²

Now, this expression can be further factorized as a difference of squares:

⇒ (a² - 3b²)² - 9² = (a² - 3b² + 9)(a² - 3b² - 9)

`\hrulefill`

Additional Information:

Difference of squares: The difference of squares is a special pattern in algebra, given by a² - b². It can be factored as (a + b)(a - b). In the given expressions, we have a² - 4b² and a² - 3b², which can be factored using this pattern.

• a² + b² = (a + b)(a - b)

Perfect square: A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be written as 4² (4 · 4 = 16). In the given expressions, 256 and 81 are perfect squares.

Factoring: Factoring is the process of expressing an algebraic expression as a product of its factors. Factors are the quantities that multiply together to give the original expression. In the given problem, we are using factorization to break down the expressions into simpler forms.