Question
1) Domain Restrictions for the Matching Polynomials:
a) Since the expression x/(x - 4) can never have a zero denominator that implies the domain restriction. Thus, x - 4 ≠ 0; x ≠ 4 implies the domain restriction.
b) The given equation, (x-8)/(x^2(x+3)), contains two things in the denominator that can never be zero. We can take x^2 ≠ 0 first and get x ≠ 0. Then we can take x + 3 ≠ 0 and get x ≠ - 3. Overall, we have x ≠ 0 and x ≠ -3. d) From the restriction given by the expression 5 - x ≠ 0, we will get x ≠ 5.
d) Factor the equation x^2 - 5x + 6 into x - 2 * x - 3. Thus, setting each term to not equal to zero gives x - 2 ≠ 0 and x - 3 ≠ 0. Hence, x = 2 and x = 3.
**2) Rational Expression Operations - Simplified:**
1) To add 5/(x+3) + 2/(x-2, find the common denominator of x+3*x-2:
(5x - 10 + 2x + 6)/((x + 3)(x - 2)) = (7x - 4)/((x + 3)(x - 2)) = (5(x - 2) + 2(x + 3)) or 5x - 10 + 2x + 6)
We firstly add the denominator for the following:
(x - 5)(x + 2) = x^2 - 3x - 10
We now simplify :
(x - 5)/((x - 5)(x + 2)) = 1/(x + 2) (x ≠ 5)
3) We can factor the numerator and the denominator of the expression (x^2 - 4)/(x^2 + 4x - 12):
(x - 2)(x + 2) = x^2 - 4
(x + 6)(x - 2) = x^2 + 4x - 12
Thus,
((x-2)(x+2))/((x + 6)(x-2)) = (x + 2)/(x + 6) (x ≠ 2)
4) We begin with simplifying the denominator of 3x^2/(3x - 6x):
3x - 6x = -3x ≠ 0
We show that:
3x^2 / -3x = -x
5) To x/5 + (x^2 + 2x - 8)/4 = 1 may be solved to obtain the common denominator. Twenty is the least common multiple of five and four. After dividing every term by 20, we get:
x^2 + 2x - 8)/4) + 20(x/5) = 20(1)
These reduce to:
20 = 4x + 5(x^2 + 2x - 8)
Distributing yields:
20 = 4x + 5x^2 + 10x - 40
And collecting terms we get,
20 - 40 - 5x2 + 14x = 0
Thus,
5x2 + 14x - 60 = 0
As a=5, b=14, and c=-60, we use quadratic formula.
x = – b ± (14^2 – 4 * 5 * (-60))^(1/2))/(2a).
The discriminant gives us:
1396 = 196 + 1200 = 14^2 - 4 * 5 * (-60)
Thus, we have:
Thus, the solutions x = (-14 ± −1396)/10 = (-14 ± 2−349)/10 = (-7 ± −349)/5 may be obtained.
Solutions:
either x = (-7-√349)/5 or x = (-7+√349)/5.
x/5 + (x^2 + 2x - 8)/4 = 1 can be solved for the common denominator. The least common multiple of five and four is twenty. After dividing every term by 20, we get:
x^2 + 2x - 8)/4) + 20(x/5) = 20(1)
These reduce to:
20=4x+5(x^2 + 2x - 8)
Distributing gives,
20 = 4x + 5x2 +10x −40
And collecting terms we have,
20 - 40 - 5x2 + 14x = 0
So,
5x2 + 14x −60 = 0
We will use the quadratic formula since a=5, b=14 and c=−60.
x = −b±(14^2 − 4 * 5 * (−60))^(1/2))/(2a).
The discriminant gives,
1396 = 196 + 1200 = 14^2 - 4 * 5 * (-60)
Thus we get,
Therefore, x = (-14 ± −√1396)/10 = (-14 ± 2−√349)/10 = (-7 ± −√349)/5 can be obtained.
Answers:
either x = (-7-−√349)/5 or x = (-7+√349)/5. epand this
a) Since the expression x/(x - 4) can never have a zero denominator that implies the domain restriction. Thus, x - 4 ≠ 0; x ≠ 4 implies the domain restriction.
b) The given equation, (x-8)/(x^2(x+3)), contains two things in the denominator that can never be zero. We can take x^2 ≠ 0 first and get x ≠ 0. Then we can take x + 3 ≠ 0 and get x ≠ - 3. Overall, we have x ≠ 0 and x ≠ -3. d) From the restriction given by the expression 5 - x ≠ 0, we will get x ≠ 5.
d) Factor the equation x^2 - 5x + 6 into x - 2 * x - 3. Thus, setting each term to not equal to zero gives x - 2 ≠ 0 and x - 3 ≠ 0. Hence, x = 2 and x = 3.
**2) Rational Expression Operations - Simplified:**
1) To add 5/(x+3) + 2/(x-2, find the common denominator of x+3*x-2:
(5x - 10 + 2x + 6)/((x + 3)(x - 2)) = (7x - 4)/((x + 3)(x - 2)) = (5(x - 2) + 2(x + 3)) or 5x - 10 + 2x + 6)
We firstly add the denominator for the following:
(x - 5)(x + 2) = x^2 - 3x - 10
We now simplify :
(x - 5)/((x - 5)(x + 2)) = 1/(x + 2) (x ≠ 5)
3) We can factor the numerator and the denominator of the expression (x^2 - 4)/(x^2 + 4x - 12):
(x - 2)(x + 2) = x^2 - 4
(x + 6)(x - 2) = x^2 + 4x - 12
Thus,
((x-2)(x+2))/((x + 6)(x-2)) = (x + 2)/(x + 6) (x ≠ 2)
4) We begin with simplifying the denominator of 3x^2/(3x - 6x):
3x - 6x = -3x ≠ 0
We show that:
3x^2 / -3x = -x
5) To x/5 + (x^2 + 2x - 8)/4 = 1 may be solved to obtain the common denominator. Twenty is the least common multiple of five and four. After dividing every term by 20, we get:
x^2 + 2x - 8)/4) + 20(x/5) = 20(1)
These reduce to:
20 = 4x + 5(x^2 + 2x - 8)
Distributing yields:
20 = 4x + 5x^2 + 10x - 40
And collecting terms we get,
20 - 40 - 5x2 + 14x = 0
Thus,
5x2 + 14x - 60 = 0
As a=5, b=14, and c=-60, we use quadratic formula.
x = – b ± (14^2 – 4 * 5 * (-60))^(1/2))/(2a).
The discriminant gives us:
1396 = 196 + 1200 = 14^2 - 4 * 5 * (-60)
Thus, we have:
Thus, the solutions x = (-14 ± −1396)/10 = (-14 ± 2−349)/10 = (-7 ± −349)/5 may be obtained.
Solutions:
either x = (-7-√349)/5 or x = (-7+√349)/5.
x/5 + (x^2 + 2x - 8)/4 = 1 can be solved for the common denominator. The least common multiple of five and four is twenty. After dividing every term by 20, we get:
x^2 + 2x - 8)/4) + 20(x/5) = 20(1)
These reduce to:
20=4x+5(x^2 + 2x - 8)
Distributing gives,
20 = 4x + 5x2 +10x −40
And collecting terms we have,
20 - 40 - 5x2 + 14x = 0
So,
5x2 + 14x −60 = 0
We will use the quadratic formula since a=5, b=14 and c=−60.
x = −b±(14^2 − 4 * 5 * (−60))^(1/2))/(2a).
The discriminant gives,
1396 = 196 + 1200 = 14^2 - 4 * 5 * (-60)
Thus we get,
Therefore, x = (-14 ± −√1396)/10 = (-14 ± 2−√349)/10 = (-7 ± −√349)/5 can be obtained.
Answers:
either x = (-7-−√349)/5 or x = (-7+√349)/5. epand this
Answers
GPT-4o mini
Sure, let's break down and expand each part of your request step by step.
### 1) Domain Restrictions for Matching Polynomials:
a) **Expression**: \(\frac{x}{x - 4}\)
- **Domain Restriction**: The denominator cannot be zero. Therefore:
\[
x - 4 \neq 0 \implies x \neq 4
\]
b) **Expression**: \(\frac{x - 8}{x^2(x + 3)}\)
- **Domain Restrictions**:
- \(x^2 \neq 0\) implies \(x \neq 0\)
- \(x + 3 \neq 0\) implies \(x \neq -3\)
Overall restrictions:
\[
x \neq 0 \quad \text{and} \quad x \neq -3
\]
c) **Expression**: \(5 - x\)
- **Domain Restriction**:
\[
5 - x \neq 0 \implies x \neq 5
\]
d) **Expression**: \(x^2 - 5x + 6\)
- **Factored Form**: \((x - 2)(x - 3)\)
- **Domain Restrictions**:
- \(x - 2 \neq 0\) implies \(x \neq 2\)
- \(x - 3 \neq 0\) implies \(x \neq 3\)
### 2) Rational Expression Operations - Simplified:
1. **Add**: \(\frac{5}{x+3} + \frac{2}{x-2}\)
- **Common Denominator**: \((x + 3)(x - 2)\)
- **Combined Expression**:
\[
\frac{5(x - 2) + 2(x + 3)}{(x + 3)(x - 2)} = \frac{5x - 10 + 2x + 6}{(x + 3)(x - 2)} = \frac{7x - 4}{(x + 3)(x - 2)}
\]
2. **Simplify**: \(\frac{x - 5}{(x - 5)(x + 2)}\)
- Canceling \(x - 5\):
\[
= \frac{1}{x + 2} \quad (x \neq 5)
\]
3. **Factor and Simplify**: \(\frac{x^2 - 4}{x^2 + 4x - 12}\)
- **Factored**:
- Numerator: \((x - 2)(x + 2)\)
- Denominator: \((x + 6)(x - 2)\)
- **Simplified Expression**:
\[
\frac{(x-2)(x+2)}{(x+6)(x-2)} = \frac{x + 2}{x + 6} \quad (x \neq 2)
\]
4. **Simplify**: \(\frac{3x^2}{3x - 6x}\)
- **Simplified Denominator**:
\[
3x - 6x = -3x \neq 0
\]
- Expression becomes:
\[
\frac{3x^2}{-3x} = -x
\]
5. **Equation**: \(\frac{x}{5} + \frac{x^2 + 2x - 8}{4} = 1\)
- **Common Denominator**: \(20\)
- **Multiply each term by 20**:
\[
20\left(\frac{x}{5}\right) + 20\left(\frac{x^2 + 2x - 8}{4}\right) = 20
\]
- This leads to:
\[
4x + 5(x^2 + 2x - 8) = 20
\]
- Distributing:
\[
4x + 5x^2 + 10x - 40 = 20
\]
- Collecting like terms yields:
\[
5x^2 + 14x - 60 = 0
\]
- **Quadratic Formula**:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \text{ where } a = 5, b = 14, c = -60
\]
- **Discriminant Calculation**:
\[
b^2 - 4ac = 14^2 - 4(5)(-60) = 196 + 1200 = 1396
\]
- Solutions:
\[
x = \frac{-14 \pm \sqrt{1396}}{10} = \frac{-14 \pm 2\sqrt{349}}{10} = \frac{-7 \pm \sqrt{349}}{5}
\]
- **Final Solutions**:
\[
x = \frac{-7 - \sqrt{349}}{5} \quad \text{or} \quad x = \frac{-7 + \sqrt{349}}{5}
\]
Thus, summarizing the final solutions, we have:
\[
\text{Either } x = \frac{-7 - \sqrt{349}}{5} \text{ or } x = \frac{-7 + \sqrt{349}}{5}.
\]
### 1) Domain Restrictions for Matching Polynomials:
a) **Expression**: \(\frac{x}{x - 4}\)
- **Domain Restriction**: The denominator cannot be zero. Therefore:
\[
x - 4 \neq 0 \implies x \neq 4
\]
b) **Expression**: \(\frac{x - 8}{x^2(x + 3)}\)
- **Domain Restrictions**:
- \(x^2 \neq 0\) implies \(x \neq 0\)
- \(x + 3 \neq 0\) implies \(x \neq -3\)
Overall restrictions:
\[
x \neq 0 \quad \text{and} \quad x \neq -3
\]
c) **Expression**: \(5 - x\)
- **Domain Restriction**:
\[
5 - x \neq 0 \implies x \neq 5
\]
d) **Expression**: \(x^2 - 5x + 6\)
- **Factored Form**: \((x - 2)(x - 3)\)
- **Domain Restrictions**:
- \(x - 2 \neq 0\) implies \(x \neq 2\)
- \(x - 3 \neq 0\) implies \(x \neq 3\)
### 2) Rational Expression Operations - Simplified:
1. **Add**: \(\frac{5}{x+3} + \frac{2}{x-2}\)
- **Common Denominator**: \((x + 3)(x - 2)\)
- **Combined Expression**:
\[
\frac{5(x - 2) + 2(x + 3)}{(x + 3)(x - 2)} = \frac{5x - 10 + 2x + 6}{(x + 3)(x - 2)} = \frac{7x - 4}{(x + 3)(x - 2)}
\]
2. **Simplify**: \(\frac{x - 5}{(x - 5)(x + 2)}\)
- Canceling \(x - 5\):
\[
= \frac{1}{x + 2} \quad (x \neq 5)
\]
3. **Factor and Simplify**: \(\frac{x^2 - 4}{x^2 + 4x - 12}\)
- **Factored**:
- Numerator: \((x - 2)(x + 2)\)
- Denominator: \((x + 6)(x - 2)\)
- **Simplified Expression**:
\[
\frac{(x-2)(x+2)}{(x+6)(x-2)} = \frac{x + 2}{x + 6} \quad (x \neq 2)
\]
4. **Simplify**: \(\frac{3x^2}{3x - 6x}\)
- **Simplified Denominator**:
\[
3x - 6x = -3x \neq 0
\]
- Expression becomes:
\[
\frac{3x^2}{-3x} = -x
\]
5. **Equation**: \(\frac{x}{5} + \frac{x^2 + 2x - 8}{4} = 1\)
- **Common Denominator**: \(20\)
- **Multiply each term by 20**:
\[
20\left(\frac{x}{5}\right) + 20\left(\frac{x^2 + 2x - 8}{4}\right) = 20
\]
- This leads to:
\[
4x + 5(x^2 + 2x - 8) = 20
\]
- Distributing:
\[
4x + 5x^2 + 10x - 40 = 20
\]
- Collecting like terms yields:
\[
5x^2 + 14x - 60 = 0
\]
- **Quadratic Formula**:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \text{ where } a = 5, b = 14, c = -60
\]
- **Discriminant Calculation**:
\[
b^2 - 4ac = 14^2 - 4(5)(-60) = 196 + 1200 = 1396
\]
- Solutions:
\[
x = \frac{-14 \pm \sqrt{1396}}{10} = \frac{-14 \pm 2\sqrt{349}}{10} = \frac{-7 \pm \sqrt{349}}{5}
\]
- **Final Solutions**:
\[
x = \frac{-7 - \sqrt{349}}{5} \quad \text{or} \quad x = \frac{-7 + \sqrt{349}}{5}
\]
Thus, summarizing the final solutions, we have:
\[
\text{Either } x = \frac{-7 - \sqrt{349}}{5} \text{ or } x = \frac{-7 + \sqrt{349}}{5}.
\]