1. $$ \left(\frac{1}{x}+2\right)\left(\frac{3}{x}-1\right) $$ 2. $$ (a b+c)\left(a^{2} b^{2}+c^{2}\right)(a b-c) $$ 3. $$ (a-2)\left(a^{3}+8\right)\left(a^{2}+2 a+4\right. $$ 4. $$ \left(a^{x}-b^{y}\right)\left(a^{x}+b^{y}\right) $$

Question

UpStudy AI · Accepted Answer

Sure, let's simplify and analyze each of the provided expressions step by step.

---

### 1. $$ \left(\frac{1}{x} + 2\right)\left(\frac{3}{x} - 1\right) $$

**Simplification:**

Multiply the two binomials using the distributive property (FOIL method):

$$
\begin{align*}
\left(\frac{1}{x} + 2\right)\left(\frac{3}{x} - 1\right) &= \frac{1}{x} \cdot \frac{3}{x} + \frac{1}{x} \cdot (-1) + 2 \cdot \frac{3}{x} + 2 \cdot (-1) \
&= \frac{3}{x^2} - \frac{1}{x} + \frac{6}{x} - 2 \
&= \frac{3}{x^2} + \left(-\frac{1}{x} + \frac{6}{x}\right) - 2 \
&= \frac{3}{x^2} + \frac{5}{x} - 2
\end{align*}
$$

**Final Simplified Form:**

$$
\frac{3}{x^2} + \frac{5}{x} - 2
$$

---

### 2. $$ (a b + c)\left(a^{2} b^{2} + c^{2}\right)(a b - c) $$

**Simplification:**

Notice that $$ (ab + c)(ab - c) $$ is a difference of squares:

$$
(ab + c)(ab - c) = (ab)^2 - c^2 = a^2b^2 - c^2
$$

Now, multiply this result by the remaining factor $$ a^{2} b^{2} + c^{2} $$:

$$
(a^2b^2 - c^2)(a^2b^2 + c^2) = (a^2b^2)^2 - (c^2)^2 = a^4b^4 - c^4
$$

**Final Simplified Form:**

$$
a^4b^4 - c^4
$$

---

### 3. $$ (a - 2)\left(a^{3} + 8\right)\left(a^{2} + 2 a + 4\right) $$

**Simplification:**

Firstly, recognize that $$ a^3 + 8 $$ is a sum of cubes:

$$
a^3 + 8 = a^3 + 2^3 = (a + 2)(a^2 - 2a + 4)
$$

Now, substitute this back into the original expression:

$$
(a - 2)(a + 2)(a^2 - 2a + 4)\left(a^{2} + 2 a + 4\right)
$$

Notice that $$ a^2 - 2a + 4 $$ and $$ a^2 + 2a + 4 $$ are quadratic terms that don’t factor further over the real numbers.

**Final Factored Form:**

$$
(a - 2)(a + 2)(a^2 - 2a + 4)(a^2 + 2a + 4)
$$

*Alternatively*, if you prefer to leave the expression partially factored:

$$
(a - 2)(a^3 + 8)(a^2 + 2a + 4)
$$

---

### 4. $$ \left(a^{x} - b^{y}\right)\left(a^{x} + b^{y}\right) $$

**Simplification:**

This follows the pattern of the difference of squares:

$$
(a^x - b^y)(a^x + b^y) = (a^x)^2 - (b^y)^2 = a^{2x} - b^{2y}
$$

**Final Simplified Form:**

$$
a^{2x} - b^{2y}
$$

---

If you have any specific questions about these simplifications or need further assistance, feel free to ask!
1. $$ \frac{3}{x^2} + \frac{5}{x} - 2 $$ 2. $$ a^4b^4 - c^4 $$ 3. $$ (a - 2)(a + 2)(a^2 - 2a + 4)(a^2 + 2a + 4) $$ 4. $$ a^{2x} - b^{2y} $$

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Algebra Questions