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Absolutely, let's walk through the process of simplifying and analyzing the expression [tex]\(\frac{x-1}{3x^2 - 11x + 10}\)[/tex] step-by-step.
### Step 1: Simplify the Expression
To simplify the given rational function, [tex]\(\frac{x-1}{3x^2 - 11x + 10}\)[/tex], we need to factor the quadratic expression in the denominator.
1. Factoring the Denominator:
The denominator is a quadratic polynomial [tex]\(3x^2 - 11x + 10\)[/tex]. We need to factor it into [tex]\( (ax + b)(cx + d) \)[/tex].
[tex]\[ 3x^2 - 11x + 10 = (3x + p)(x + q) \][/tex]
We need [tex]\(p\)[/tex] and [tex]\(q\)[/tex] such that:
- The product of the constants is [tex]\(3 \times 10 = 30\)[/tex].
- The sum of the factors that produce [tex]\(-11\)[/tex] when multiplied by their respective coefficients.
Finding the factors of [tex]\(30\)[/tex] that add up to [tex]\(-11\)[/tex] (considering signs):
- Checking pairs, [tex]\((-5\)[/tex] and [tex]\(-6\)[/tex]):
[tex]\[ 3x^2 - 11x + 10 = (3x - 5)(x - 2) \][/tex]
Verification:
- Expand [tex]\((3x - 5)(x - 2)\)[/tex] to ensure accuracy:
[tex]\[ (3x - 5)(x - 2) = 3x^2 - 6x - 5x + 10 = 3x^2 - 11x + 10 \][/tex]
Thus, the denominator factors correctly as [tex]\( (3x - 5)(x - 2) \)[/tex].
2. Rewrite the Original Expression:
With the denominator factored, our expression becomes:
[tex]\[ \frac{x - 1}{(3x - 5)(x - 2)} \][/tex]
### Step 2: Analyze the Simplified Expression
There is no common factor in the numerator and denominator, so the simplified expression is:
[tex]\[ \frac{x - 1}{(3x - 5)(x - 2)} \][/tex]
### Step 3: Determine the Domain
To identify the domain of the function, we must determine for which [tex]\(x\)[/tex]-values the denominator is non-zero, as division by zero is undefined.
Set each factor in the denominator equal to zero and solve for [tex]\(x\)[/tex]:
1. [tex]\(3x - 5 = 0\)[/tex]
[tex]\[ 3x = 5 \quad \Rightarrow \quad x = \frac{5}{3} \][/tex]
2. [tex]\(x - 2 = 0\)[/tex]
[tex]\[ x = 2 \][/tex]
These values make the denominator zero. Therefore, the function is undefined at [tex]\(x = \frac{5}{3}\)[/tex] and [tex]\(x = 2\)[/tex].
Domain of the function:
[tex]\[ x \in \mathbb{R} \setminus \left\{ \frac{5}{3}, 2 \right\} \][/tex]
### Conclusion
To summarize, the simplified and analyzed form of the function is:
[tex]\[ \frac{x - 1}{3x^2 - 11x + 10} = \frac{x - 1}{(3x - 5)(x - 2)} \][/tex]
The domain of this function excludes the points where the denominator is zero:
[tex]\[ x \in \mathbb{R} \setminus \left\{ \frac{5}{3}, 2 \right\} \][/tex]
This completes our detailed, step-by-step solution of the given mathematical expression.
### Step 1: Simplify the Expression
To simplify the given rational function, [tex]\(\frac{x-1}{3x^2 - 11x + 10}\)[/tex], we need to factor the quadratic expression in the denominator.
1. Factoring the Denominator:
The denominator is a quadratic polynomial [tex]\(3x^2 - 11x + 10\)[/tex]. We need to factor it into [tex]\( (ax + b)(cx + d) \)[/tex].
[tex]\[ 3x^2 - 11x + 10 = (3x + p)(x + q) \][/tex]
We need [tex]\(p\)[/tex] and [tex]\(q\)[/tex] such that:
- The product of the constants is [tex]\(3 \times 10 = 30\)[/tex].
- The sum of the factors that produce [tex]\(-11\)[/tex] when multiplied by their respective coefficients.
Finding the factors of [tex]\(30\)[/tex] that add up to [tex]\(-11\)[/tex] (considering signs):
- Checking pairs, [tex]\((-5\)[/tex] and [tex]\(-6\)[/tex]):
[tex]\[ 3x^2 - 11x + 10 = (3x - 5)(x - 2) \][/tex]
Verification:
- Expand [tex]\((3x - 5)(x - 2)\)[/tex] to ensure accuracy:
[tex]\[ (3x - 5)(x - 2) = 3x^2 - 6x - 5x + 10 = 3x^2 - 11x + 10 \][/tex]
Thus, the denominator factors correctly as [tex]\( (3x - 5)(x - 2) \)[/tex].
2. Rewrite the Original Expression:
With the denominator factored, our expression becomes:
[tex]\[ \frac{x - 1}{(3x - 5)(x - 2)} \][/tex]
### Step 2: Analyze the Simplified Expression
There is no common factor in the numerator and denominator, so the simplified expression is:
[tex]\[ \frac{x - 1}{(3x - 5)(x - 2)} \][/tex]
### Step 3: Determine the Domain
To identify the domain of the function, we must determine for which [tex]\(x\)[/tex]-values the denominator is non-zero, as division by zero is undefined.
Set each factor in the denominator equal to zero and solve for [tex]\(x\)[/tex]:
1. [tex]\(3x - 5 = 0\)[/tex]
[tex]\[ 3x = 5 \quad \Rightarrow \quad x = \frac{5}{3} \][/tex]
2. [tex]\(x - 2 = 0\)[/tex]
[tex]\[ x = 2 \][/tex]
These values make the denominator zero. Therefore, the function is undefined at [tex]\(x = \frac{5}{3}\)[/tex] and [tex]\(x = 2\)[/tex].
Domain of the function:
[tex]\[ x \in \mathbb{R} \setminus \left\{ \frac{5}{3}, 2 \right\} \][/tex]
### Conclusion
To summarize, the simplified and analyzed form of the function is:
[tex]\[ \frac{x - 1}{3x^2 - 11x + 10} = \frac{x - 1}{(3x - 5)(x - 2)} \][/tex]
The domain of this function excludes the points where the denominator is zero:
[tex]\[ x \in \mathbb{R} \setminus \left\{ \frac{5}{3}, 2 \right\} \][/tex]
This completes our detailed, step-by-step solution of the given mathematical expression.
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