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Sagot :
To determine which of the given rational functions are in their lowest terms, we need to factor both the numerator and the denominator and simplify each expression, if possible.
### Equation 1: [tex]\( y = \frac{2x^2 + x - 1}{x^2 - 1} \)[/tex]
Numerator: [tex]\( 2x^2 + x - 1 \)[/tex]
- Factors of [tex]\(2x^2 + x - 1\)[/tex]: [tex]\((2x - 1)(x + 1)\)[/tex]
Denominator: [tex]\( x^2 - 1 \)[/tex]
- Factors of [tex]\(x^2 - 1\)[/tex]: [tex]\((x - 1)(x + 1)\)[/tex]
Thus, the expression simplifies to:
[tex]\[ y = \frac{(2x - 1)(x + 1)}{(x - 1)(x + 1)} \][/tex]
The [tex]\( (x + 1) \)[/tex] terms cancel out, leaving:
[tex]\[ y = \frac{2x - 1}{x - 1} \][/tex]
This is now in its simplest form.
### Equation 2: [tex]\( y = \frac{x^2 + 11x + 18}{x^2 - 11x + 18} \)[/tex]
Numerator: [tex]\( x^2 + 11x + 18 \)[/tex]
- Factors of [tex]\(x^2 + 11x + 18\)[/tex]: [tex]\((x + 9)(x + 2)\)[/tex]
Denominator: [tex]\( x^2 - 11x + 18 \)[/tex]
- Factors of [tex]\(x^2 - 11x + 18\)[/tex]: [tex]\((x - 2)(x - 9)\)[/tex]
There are no common factors to simplify, so the expression remains as it is.
### Equation 3: [tex]\( y = \frac{2x^2 + 6x - 8}{2x^2 + 4x - 6} \)[/tex]
Numerator: [tex]\( 2x^2 + 6x - 8 \)[/tex]
- Factors of [tex]\(2x^2 + 6x - 8\)[/tex]: [tex]\((2x - 2)(x + 4)\)[/tex] (factors to [tex]\(2(x + 4)(x - 1)\)[/tex])
Denominator: [tex]\( 2x^2 + 4x - 6 \)[/tex]
- Factors of [tex]\(2x^2 + 4x - 6\)[/tex]: [tex]\((2x - 2)(x + 3)\)[/tex] (factors to [tex]\(2(x + 3)(x - 1)\)[/tex])
Thus, the expression simplifies to:
[tex]\[ y = \frac{2(x + 4)(x - 1)}{2(x + 3)(x - 1)} \][/tex]
The [tex]\(2\)[/tex] and [tex]\( (x - 1) \)[/tex] terms cancel out, leaving:
[tex]\[ y = \frac{x + 4}{x + 3} \][/tex]
This is now in its simplest form.
### Equation 4: [tex]\( y = \frac{x^2 + x - 56}{x^2 - 12x + 35} \)[/tex]
Numerator: [tex]\( x^2 + x - 56 \)[/tex]
- Factors of [tex]\(x^2 + x - 56\)[/tex]: [tex]\((x + 8)(x - 7)\)[/tex]
Denominator: [tex]\( x^2 - 12x + 35 \)[/tex]
- Factors of [tex]\(x^2 - 12x + 35\)[/tex]: [tex]\((x - 5)(x - 7)\)[/tex]
Thus, the expression simplifies to:
[tex]\[ y = \frac{(x + 8)(x - 7)}{(x - 5)(x - 7)} \][/tex]
The [tex]\( (x - 7) \)[/tex] terms cancel out, leaving:
[tex]\[ y = \frac{x + 8}{x - 5} \][/tex]
This is now in its simplest form.
### Equation 5: [tex]\( y = \frac{2x^2 - 5x + 3}{x^2 + 4x + 3} \)[/tex]
Numerator: [tex]\( 2x^2 - 5x + 3 \)[/tex]
- Factors of [tex]\(2x^2 - 5x + 3\)[/tex]: [tex]\((2x - 3)(x - 1)\)[/tex]
Denominator: [tex]\( x^2 + 4x + 3 \)[/tex]
- Factors of [tex]\(x^2 + 4x + 3\)[/tex]: [tex]\((x + 3)(x + 1)\)[/tex]
There are no common factors to simplify, so the expression remains as it is.
### Summary
The equations in their lowest terms are:
1. [tex]\( y = \frac{2x^2 + x - 1}{x^2 - 1} \)[/tex]
2. [tex]\( y = \frac{2x^2 + 6x - 8}{2x^2 + 4x - 6} \)[/tex]
4. [tex]\( y = \frac{x^2 + x - 56}{x^2 - 12x + 35} \)[/tex]
Therefore, the correct answers are: [tex]\( \boxed{1, 3, 4} \)[/tex].
### Equation 1: [tex]\( y = \frac{2x^2 + x - 1}{x^2 - 1} \)[/tex]
Numerator: [tex]\( 2x^2 + x - 1 \)[/tex]
- Factors of [tex]\(2x^2 + x - 1\)[/tex]: [tex]\((2x - 1)(x + 1)\)[/tex]
Denominator: [tex]\( x^2 - 1 \)[/tex]
- Factors of [tex]\(x^2 - 1\)[/tex]: [tex]\((x - 1)(x + 1)\)[/tex]
Thus, the expression simplifies to:
[tex]\[ y = \frac{(2x - 1)(x + 1)}{(x - 1)(x + 1)} \][/tex]
The [tex]\( (x + 1) \)[/tex] terms cancel out, leaving:
[tex]\[ y = \frac{2x - 1}{x - 1} \][/tex]
This is now in its simplest form.
### Equation 2: [tex]\( y = \frac{x^2 + 11x + 18}{x^2 - 11x + 18} \)[/tex]
Numerator: [tex]\( x^2 + 11x + 18 \)[/tex]
- Factors of [tex]\(x^2 + 11x + 18\)[/tex]: [tex]\((x + 9)(x + 2)\)[/tex]
Denominator: [tex]\( x^2 - 11x + 18 \)[/tex]
- Factors of [tex]\(x^2 - 11x + 18\)[/tex]: [tex]\((x - 2)(x - 9)\)[/tex]
There are no common factors to simplify, so the expression remains as it is.
### Equation 3: [tex]\( y = \frac{2x^2 + 6x - 8}{2x^2 + 4x - 6} \)[/tex]
Numerator: [tex]\( 2x^2 + 6x - 8 \)[/tex]
- Factors of [tex]\(2x^2 + 6x - 8\)[/tex]: [tex]\((2x - 2)(x + 4)\)[/tex] (factors to [tex]\(2(x + 4)(x - 1)\)[/tex])
Denominator: [tex]\( 2x^2 + 4x - 6 \)[/tex]
- Factors of [tex]\(2x^2 + 4x - 6\)[/tex]: [tex]\((2x - 2)(x + 3)\)[/tex] (factors to [tex]\(2(x + 3)(x - 1)\)[/tex])
Thus, the expression simplifies to:
[tex]\[ y = \frac{2(x + 4)(x - 1)}{2(x + 3)(x - 1)} \][/tex]
The [tex]\(2\)[/tex] and [tex]\( (x - 1) \)[/tex] terms cancel out, leaving:
[tex]\[ y = \frac{x + 4}{x + 3} \][/tex]
This is now in its simplest form.
### Equation 4: [tex]\( y = \frac{x^2 + x - 56}{x^2 - 12x + 35} \)[/tex]
Numerator: [tex]\( x^2 + x - 56 \)[/tex]
- Factors of [tex]\(x^2 + x - 56\)[/tex]: [tex]\((x + 8)(x - 7)\)[/tex]
Denominator: [tex]\( x^2 - 12x + 35 \)[/tex]
- Factors of [tex]\(x^2 - 12x + 35\)[/tex]: [tex]\((x - 5)(x - 7)\)[/tex]
Thus, the expression simplifies to:
[tex]\[ y = \frac{(x + 8)(x - 7)}{(x - 5)(x - 7)} \][/tex]
The [tex]\( (x - 7) \)[/tex] terms cancel out, leaving:
[tex]\[ y = \frac{x + 8}{x - 5} \][/tex]
This is now in its simplest form.
### Equation 5: [tex]\( y = \frac{2x^2 - 5x + 3}{x^2 + 4x + 3} \)[/tex]
Numerator: [tex]\( 2x^2 - 5x + 3 \)[/tex]
- Factors of [tex]\(2x^2 - 5x + 3\)[/tex]: [tex]\((2x - 3)(x - 1)\)[/tex]
Denominator: [tex]\( x^2 + 4x + 3 \)[/tex]
- Factors of [tex]\(x^2 + 4x + 3\)[/tex]: [tex]\((x + 3)(x + 1)\)[/tex]
There are no common factors to simplify, so the expression remains as it is.
### Summary
The equations in their lowest terms are:
1. [tex]\( y = \frac{2x^2 + x - 1}{x^2 - 1} \)[/tex]
2. [tex]\( y = \frac{2x^2 + 6x - 8}{2x^2 + 4x - 6} \)[/tex]
4. [tex]\( y = \frac{x^2 + x - 56}{x^2 - 12x + 35} \)[/tex]
Therefore, the correct answers are: [tex]\( \boxed{1, 3, 4} \)[/tex].
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