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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

1 answer

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: \[ \text{Either } x = \frac{-7 - \sqrt{349}}{5} \text{ or } x = \frac{-7 + \sqrt{349}}{5}. \]