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

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UpStudy AI · Accepted Answer

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!
a) $$ x^{2} - 4 = (x - 2)(x + 2) $$ b) $$ x^{4} - 16 = (x - 2)(x + 2)(x^{2} + 4) $$ c) $$ 2a^{2} - 8b^{2} = 2(a - 2b)(a + 2b) $$ d) $$ 36p^{2}r - 16q^{2}r = 4r(3p - 2q)(3p + 2q) $$ e) $$ \frac{2}{x^{2}} - \frac{8}{y^{7}} = \frac{2}{x^{2}}\left(1 - \frac{4x^{2}}{y^{7}}\right) $$ f) $$ \frac{x^{2}}{9} - \frac{y^{2}}{16} = \left(\frac{x}{3} - \frac{y}{4}\right)\left(\frac{x}{3} + \frac{y}{4}\right) $$ g) $$ (x - 3)^{2} - 36 = (x - 9)(x + 3) $$ h) $$ 2(x + 2)^{2} - 8 = 2x(x + 4) $$ i) $$ 16x^{2} + 4y^{2} = 4(4x^{2} + y^{2}) $$

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