(2) $$ x^{2}-4 $$ (5) $$ \quad 9 x^{2}-25 y^{2} $$ (8) $$ a^{10}-b^{6} $$

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UpStudy AI · Accepted Answer

Certainly! Let's factor each of the given expressions step by step.

### (2) $$ x^{2} - 4 $$

**Type of Expression:** Difference of Squares

The expression can be written as:
$$
x^{2} - 2^{2}
$$
Using the difference of squares formula $$ a^{2} - b^{2} = (a - b)(a + b) $$:
$$
x^{2} - 4 = (x - 2)(x + 2)
$$

### (5) $$ 9x^{2} - 25y^{2} $$

**Type of Expression:** Difference of Squares

First, recognize that both terms are perfect squares:
$$
9x^{2} = (3x)^{2} \quad \text{and} \quad 25y^{2} = (5y)^{2}
$$
So, the expression becomes:
$$
(3x)^{2} - (5y)^{2}
$$
Applying the difference of squares formula:
$$
9x^{2} - 25y^{2} = (3x - 5y)(3x + 5y)
$$

### (8) $$ a^{10} - b^{6} $$

**Type of Expression:** Difference of Powers (can be treated as a difference of squares)

First, express the terms as squares:
$$
a^{10} = (a^{5})^{2} \quad \text{and} \quad b^{6} = (b^{3})^{2}
$$
Thus:
$$
a^{10} - b^{6} = (a^{5})^{2} - (b^{3})^{2}
$$
Applying the difference of squares formula:
$$
a^{10} - b^{6} = (a^{5} - b^{3})(a^{5} + b^{3})
$$
This is the fully factored form over the integers. Further factoring may not be possible without additional constraints or information.

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**Summary of Factored Forms:**

1. $$x^{2} - 4 = (x - 2)(x + 2)$$
2. $$9x^{2} - 25y^{2} = (3x - 5y)(3x + 5y)$$
3. $$a^{10} - b^{6} = (a^{5} - b^{3})(a^{5} + b^{3})$$
Here are the factored forms of the given expressions: 1. $$ x^{2} - 4 = (x - 2)(x + 2) $$ 2. $$ 9x^{2} - 25y^{2} = (3x - 5y)(3x + 5y) $$ 3. $$ a^{10} - b^{6} = (a^{5} - b^{3})(a^{5} + b^{3}) $$

(2) \( x^{2}-4 \) (5) \( \quad 9 x^{2}-25 y^{2} \) (8) \( a^{10}-b^{6} \)

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