Question
Write the polynomial expression in simplest form:
2x^2 -32/x-4 * 10x +20/5x^2- 20
The simplified form is
(Keep answer in factored form.)
What are the restrictions on the variable?
(Type only the numbers and separate your answers with commas, if needed.)
2x^2 -32/x-4 * 10x +20/5x^2- 20
The simplified form is
(Keep answer in factored form.)
What are the restrictions on the variable?
(Type only the numbers and separate your answers with commas, if needed.)
Answers
GPT-4o mini
To simplify the polynomial expression
\[
\frac{2x^2 - 32}{x - 4} \cdot \frac{10x + 20}{5x^2 - 20},
\]
let's start by factoring each part of the expression.
**Step 1: Factor the terms.**
1. Factor \(2x^2 - 32\):
\[
2(x^2 - 16) = 2(x - 4)(x + 4).
\]
2. Factor \(x - 4\):
This remains as \(x - 4\).
3. Factor \(10x + 20\):
\[
10(x + 2).
\]
4. Factor \(5x^2 - 20\):
\[
5(x^2 - 4) = 5(x - 2)(x + 2).
\]
**Step 2: Rewrite the expression using the factored forms.**
Now substitute the factored expressions back into the original expression:
\[
\frac{2(x - 4)(x + 4)}{x - 4} \cdot \frac{10(x + 2)}{5(x - 2)(x + 2)}.
\]
**Step 3: Cancel common terms.**
- The \(x - 4\) in the numerator and denominator cancels:
\[
2(x + 4) \cdot \frac{10(x + 2)}{5(x - 2)(x + 2)}.
\]
- The \(x + 2\) in the numerator and denominator cancels:
\[
2(x + 4) \cdot \frac{10}{5(x - 2)}.
\]
This simplifies to:
\[
\frac{2 \cdot 10 (x + 4)}{5(x - 2)} = \frac{20(x + 4)}{5(x - 2)}.
\]
**Step 4: Further simplify and factor.**
\[
\frac{20(x + 4)}{5(x - 2)} = \frac{4(x + 4)}{x - 2}.
\]
Thus, the final simplified form in factored form is:
\[
\frac{4(x + 4)}{x - 2}.
\]
**Restrictions on the variable:**
To determine restrictions on the variable \(x\), we need to identify values for which the original expression is undefined. This occurs whenever the denominator equals zero:
1. From \(x - 4\):
\[
x - 4 = 0 \Rightarrow x = 4.
\]
2. From \(5(x - 2)(x + 2)\):
- \(x - 2 = 0 \Rightarrow x = 2\),
- \(x + 2 = 0 \Rightarrow x = -2\).
So the restrictions on \(x\) are:
\[
x = 4, 2, -2.
\]
**Final Answer:**
The simplified expression is:
\[
\frac{4(x + 4)}{x - 2}.
\]
The restrictions on the variable \(x\) are:
\[
4, 2, -2.
\]
\[
\frac{2x^2 - 32}{x - 4} \cdot \frac{10x + 20}{5x^2 - 20},
\]
let's start by factoring each part of the expression.
**Step 1: Factor the terms.**
1. Factor \(2x^2 - 32\):
\[
2(x^2 - 16) = 2(x - 4)(x + 4).
\]
2. Factor \(x - 4\):
This remains as \(x - 4\).
3. Factor \(10x + 20\):
\[
10(x + 2).
\]
4. Factor \(5x^2 - 20\):
\[
5(x^2 - 4) = 5(x - 2)(x + 2).
\]
**Step 2: Rewrite the expression using the factored forms.**
Now substitute the factored expressions back into the original expression:
\[
\frac{2(x - 4)(x + 4)}{x - 4} \cdot \frac{10(x + 2)}{5(x - 2)(x + 2)}.
\]
**Step 3: Cancel common terms.**
- The \(x - 4\) in the numerator and denominator cancels:
\[
2(x + 4) \cdot \frac{10(x + 2)}{5(x - 2)(x + 2)}.
\]
- The \(x + 2\) in the numerator and denominator cancels:
\[
2(x + 4) \cdot \frac{10}{5(x - 2)}.
\]
This simplifies to:
\[
\frac{2 \cdot 10 (x + 4)}{5(x - 2)} = \frac{20(x + 4)}{5(x - 2)}.
\]
**Step 4: Further simplify and factor.**
\[
\frac{20(x + 4)}{5(x - 2)} = \frac{4(x + 4)}{x - 2}.
\]
Thus, the final simplified form in factored form is:
\[
\frac{4(x + 4)}{x - 2}.
\]
**Restrictions on the variable:**
To determine restrictions on the variable \(x\), we need to identify values for which the original expression is undefined. This occurs whenever the denominator equals zero:
1. From \(x - 4\):
\[
x - 4 = 0 \Rightarrow x = 4.
\]
2. From \(5(x - 2)(x + 2)\):
- \(x - 2 = 0 \Rightarrow x = 2\),
- \(x + 2 = 0 \Rightarrow x = -2\).
So the restrictions on \(x\) are:
\[
x = 4, 2, -2.
\]
**Final Answer:**
The simplified expression is:
\[
\frac{4(x + 4)}{x - 2}.
\]
The restrictions on the variable \(x\) are:
\[
4, 2, -2.
\]