(2) \( x^{2}-4 \) (5) \( 9 x^{2}-25 y^{2} \) (8) \( a^{10}-b^{6} \) (11) \( y^{8}-1 \) (14) \( \frac{a^{4}}{16}-\frac{1}{25} b^{2} \)

Sure, let's factor each of the given expressions step by step. --- ### **(2) \( x^{2} - 4 \)** **Type of Expression:** Difference of Squares **Factorization:** \[ x^{2} - 4 = x^{2} - 2^{2} = (x - 2)(x + 2) \] --- ### **(5) \( 9x^{2} - 25y^{2} \)** **Type of Expression:** Difference of Squares **Factorization:** \[ 9x^{2} - 25y^{2} = (3x)^{2} - (5y)^{2} = (3x - 5y)(3x + 5y) \] --- ### **(8) \( a^{10} - b^{6} \)** **Type of Expression:** Difference of Squares (with higher exponents) **Factorization:** 1. **First Step: Recognize as a Difference of Squares** \[ a^{10} - b^{6} = (a^{5})^{2} - (b^{3})^{2} = (a^{5} - b^{3})(a^{5} + b^{3}) \] 2. **Further Factoring (if possible):** - \( a^{5} - b^{3} \) and \( a^{5} + b^{3} \) don't factor nicely over the real numbers beyond this point. **Final Factorization:** \[ a^{10} - b^{6} = (a^{5} - b^{3})(a^{5} + b^{3}) \] --- ### **(11) \( y^{8} - 1 \)** **Type of Expression:** Difference of Squares (repeatedly applied) **Factorization:** 1. **First Step:** \[ y^{8} - 1 = (y^{4})^{2} - 1^{2} = (y^{4} - 1)(y^{4} + 1) \] 2. **Factor \( y^{4} - 1 \) Further:** \[ y^{4} - 1 = (y^{2})^{2} - 1^{2} = (y^{2} - 1)(y^{2} + 1) = (y - 1)(y + 1)(y^{2} + 1) \] 3. **Combine All Factors:** \[ y^{8} - 1 = (y - 1)(y + 1)(y^{2} + 1)(y^{4} + 1) \] *Note:* \( y^{4} + 1 \) can be factored further using complex numbers, but it's typically left as is in real factorization. --- ### **(14) \( \frac{a^{4}}{16} - \frac{b^{2}}{25} \)** **Type of Expression:** Difference of Squares with Fractional Coefficients **Factorization:** 1. **Rewrite the Expression as Squares:** \[ \frac{a^{4}}{16} - \frac{b^{2}}{25} = \left(\frac{a^{2}}{4}\right)^{2} - \left(\frac{b}{5}\right)^{2} \] 2. **Apply Difference of Squares Formula:** \[ \left(\frac{a^{2}}{4}\right)^{2} - \left(\frac{b}{5}\right)^{2} = \left(\frac{a^{2}}{4} - \frac{b}{5}\right)\left(\frac{a^{2}}{4} + \frac{b}{5}\right) \] **Final Factorization:** \[ \frac{a^{4}}{16} - \frac{b^{2}}{25} = \left(\frac{a^{2}}{4} - \frac{b}{5}\right)\left(\frac{a^{2}}{4} + \frac{b}{5}\right) \] --- Each of these factorizations utilizes the **Difference of Squares** method, which states that \( A^{2} - B^{2} = (A - B)(A + B) \). This method is fundamental in algebra for simplifying and solving various types of equations.