Factorbe the following: Ca) \( x^{2}-4 \) b) \( x^{4}-16 \) c) \( 2 a^{2}-8 b^{2} \) d) \( 36 p^{2} r-16 q^{2} r \) e) \( \left(\frac{2}{x^{2}}-\frac{8}{y^{7}}\right) \quad \) Quodratic f) \( \left(\frac{x^{2}}{9}-\frac{y^{2}}{16}\right) \quad \) equation g) \( (x-3)^{2}-36 \) h) \( 2(x+2)^{2}-8 \) i) \( 16 x^{2}+4 y^{2} \) a

Certainly! Let's factor each of the given expressions step by step. --- ### **a) \( x^{2} - 4 \)** **Factorization:** This is a **difference of squares**, which follows the formula \( a^2 - b^2 = (a - b)(a + b) \). \[ x^{2} - 4 = x^{2} - 2^{2} = (x - 2)(x + 2) \] --- ### **b) \( x^{4} - 16 \)** **Factorization:** First, recognize that \( x^{4} - 16 \) is a **difference of squares**: \[ x^{4} - 16 = (x^{2})^{2} - 4^{2} = (x^{2} - 4)(x^{2} + 4) \] The term \( x^{2} - 4 \) can be further factored as a difference of squares: \[ x^{2} - 4 = (x - 2)(x + 2) \] So, the complete factorization is: \[ x^{4} - 16 = (x - 2)(x + 2)(x^{2} + 4) \] --- ### **c) \( 2a^{2} - 8b^{2} \)** **Factorization:** 1. **Factor out the greatest common factor (GCF), which is 2:** \[ 2a^{2} - 8b^{2} = 2(a^{2} - 4b^{2}) \] 2. **Factor the expression inside the parentheses as a difference of squares:** \[ a^{2} - 4b^{2} = (a)^{2} - (2b)^{2} = (a - 2b)(a + 2b) \] 3. **Combine the factors:** \[ 2(a^{2} - 4b^{2}) = 2(a - 2b)(a + 2b) \] --- ### **d) \( 36p^{2}r - 16q^{2}r \)** **Factorization:** 1. **Factor out the GCF, which is \( 4r \):** \[ 36p^{2}r - 16q^{2}r = 4r(9p^{2} - 4q^{2}) \] 2. **Factor the expression inside the parentheses as a difference of squares:** \[ 9p^{2} - 4q^{2} = (3p)^{2} - (2q)^{2} = (3p - 2q)(3p + 2q) \] 3. **Combine the factors:** \[ 4r(9p^{2} - 4q^{2}) = 4r(3p - 2q)(3p + 2q) \] --- ### **e) \( \left(\frac{2}{x^{2}} - \frac{8}{y^{7}}\right) \)** **Factorization:** 1. **Factor out the GCF, which is \( \frac{2}{x^{2}} \):** \[ \frac{2}{x^{2}} - \frac{8}{y^{7}} = \frac{2}{x^{2}}\left(1 - \frac{4x^{2}}{y^{7}}\right) \] 2. **Alternatively, if you prefer factoring numerators:** \[ \frac{2}{x^{2}} - \frac{8}{y^{7}} = 2\left(\frac{1}{x^{2}} - \frac{4}{y^{7}}\right) \] *Note:* Since this expression involves variables in the denominator and exponents that do not form a standard polynomial structure, it's primarily factored by extracting the common factor. --- ### **f) \( \left(\frac{x^{2}}{9} - \frac{y^{2}}{16}\right) \)** **Factorization:** This is a **difference of squares**: \[ \frac{x^{2}}{9} - \frac{y^{2}}{16} = \left(\frac{x}{3}\right)^{2} - \left(\frac{y}{4}\right)^{2} \] Using the difference of squares formula: \[ \left(\frac{x}{3}\right)^{2} - \left(\frac{y}{4}\right)^{2} = \left(\frac{x}{3} - \frac{y}{4}\right)\left(\frac{x}{3} + \frac{y}{4}\right) \] Alternatively, you can write: \[ \frac{x^{2}}{9} - \frac{y^{2}}{16} = \frac{4x^2 - 9y^2}{36} = \frac{(2x - 3y)(2x + 3y)}{36} \] --- ### **g) \( (x - 3)^{2} - 36 \)** **Factorization:** 1. **Recognize the expression as a difference of squares:** \[ (x - 3)^{2} - 6^{2} = \left( x - 3 - 6 \right)\left( x - 3 + 6 \right) \] 2. **Simplify the terms:** \[ (x - 9)(x + 3) \] --- ### **h) \( 2(x + 2)^{2} - 8 \)** **Factorization:** 1. **Factor out the GCF, which is 2:** \[ 2(x + 2)^{2} - 8 = 2\left( (x + 2)^{2} - 4 \right) \] 2. **Recognize the expression inside the parentheses as a difference of squares:** \[ (x + 2)^{2} - 2^{2} = (x + 2 - 2)(x + 2 + 2) = x(x + 4) \] 3. **Combine the factors:** \[ 2\left( (x + 2)^{2} - 4 \right) = 2x(x + 4) \] --- ### **i) \( 16x^{2} + 4y^{2} \)** **Factorization:** 1. **Factor out the GCF, which is 4:** \[ 16x^{2} + 4y^{2} = 4(4x^{2} + y^{2}) \] 2. **Note:** The expression \( 4x^{2} + y^{2} \) is a **sum of squares** and cannot be factored further over the set of real numbers. - *Over complex numbers, it can be factored as:* \[ 4x^{2} + y^{2} = (2x - iy)(2x + iy) \] where \( i = \sqrt{-1} \). **Final Factorization:** \[ 16x^{2} + 4y^{2} = 4(4x^{2} + y^{2}) \] --- Feel free to ask if you need further clarification on any of these factorizations!