How do you reduce the rational expression (x^2 + 5x - 6) / (x - 1)?

Factor the numerator into (x - 1)(x + 6). Since x ≠ 1, the (x - 1) terms cancel out, leaving x + 6.

How do you simplify (5 - x) / (2x - 10)?

Factor the denominator to 2(x - 5). Notice that (5 - x) is the negative of (x - 5). By factoring out -1, we get -(x - 5) / 2(x - 5), which simplifies to -1/2.

Reducing Rational Expressions: Practice & Solutions

Mastering rational expressions involves two critical steps: factoring the numerator and denominator completely, and dividing out common factors. Detailed solutions are provided below to help you identify common algebraic traps.

Practice Questions

Question 1

For \( x \ne 1 \), simplify: \[ \dfrac{x^2 + 5x - 6}{x - 1} \]

Show Detailed Solution

1. Factor the numerator: \(x^2 + 5x - 6\) becomes \((x - 1)(x + 6)\).

2. The expression is now \(\frac{(x - 1)(x + 6)}{x - 1}\).

3. Since \(x \ne 1\), the term \((x-1)\) is not zero. We can cancel it: \(\mathbf{x + 6}\).

Question 2

Simplify for \( x \ne 5 \): \[ \dfrac{5 - x}{2x - 10} \]

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1. Factor the denominator: \(2x - 10 = 2(x - 5)\).

2. Notice the numerator is \((5 - x)\). This is the opposite of \((x - 5)\).

3. Rewrite the numerator as \(-(x - 5)\).

4. Expression: \(\frac{-(x - 5)}{2(x - 5)}\). Cancel \((x - 5)\) to get \(\mathbf{-1/2}\).

Question 3

For \( x \ne -4 \), simplify: \[ \dfrac{16 - x^2}{x + 4} \]

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1. Factor the numerator as a difference of squares: \(16 - x^2 = (4 - x)(4 + x)\).

2. Expression: \(\frac{(4 - x)(4 + x)}{x + 4}\).

3. Since \((4 + x)\) is the same as \((x + 4)\), they cancel out, leaving \(\mathbf{4 - x}\).

Question 4

Simplify: \[ \dfrac{x + 2}{x^2 + 2x} \]

Show Detailed Solution

1. Factor the denominator: \(x^2 + 2x = x(x + 2)\).

2. Expression: \(\frac{x + 2}{x(x + 2)}\).

3. Cancel the common factor \((x + 2)\) (with condition \(x \ne -2\)).

4. The result is \(\mathbf{1/x}\).

Question 5

For \( x \ne 3 \), simplify: \[ \dfrac{3 - x}{x^2 - x - 6} \]

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1. Factor the denominator: \(x^2 - x - 6 = (x - 3)(x + 2)\).

2. Again, the numerator \((3 - x)\) is the negative of \((x - 3)\).

3. Rewrite numerator as \(-(x - 3)\).

4. Cancel \((x - 3)\) to get \(\mathbf{-1/(x + 2)}\).

Question 6

Simplify: \[ \dfrac{x^3 - x}{x^2 - 1} \]

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1. Factor the numerator: \(x(x^2 - 1)\).

2. Expression: \(\frac{x(x^2 - 1)}{x^2 - 1}\).

3. Cancel the common binomial \((x^2 - 1)\), leaving \(\mathbf{x}\).

Question 7

Simplify: \[ \dfrac{x^2 - 4}{x^2 + 4x - 12} \]

Show Detailed Solution

1. Factor numerator: \((x - 2)(x + 2)\).

2. Factor denominator: \((x - 2)(x + 6)\).

3. Cancel the \((x - 2)\) terms, leaving \(\mathbf{(x + 2)/(x + 6)}\).

Question 8

Simplify: \[ \dfrac{x^2 + 1}{x^3 + x} \]

Show Detailed Solution

1. Factor the denominator: \(x(x^2 + 1)\).

2. Expression: \(\frac{x^2 + 1}{x(x^2 + 1)}\).

3. Cancel the \((x^2 + 1)\) factor, leaving \(\mathbf{1/x}\).

Question 9

Simplify: \[ \dfrac{x^2 + 2x - 3}{2x^2 + 3x - 5} \]

Show Detailed Solution

1. Factor numerator: \((x - 1)(x + 3)\).

2. Factor denominator: \(2x^2 + 3x - 5 = (2x + 5)(x - 1)\).

3. Cancel \((x - 1)\) to get \(\mathbf{(x + 3)/(2x + 5)}\).

Question 10

For \( x \ne 1 \), simplify: \[ \dfrac{x - 1}{(x^2 - 1)(x + 3)} \]

Show Detailed Solution

1. Expand the denominator factor: \(x^2 - 1 = (x - 1)(x + 1)\).

2. Expression: \(\frac{x - 1}{(x - 1)(x + 1)(x + 3)}\).

3. Cancel \((x - 1)\), leaving \(\mathbf{1/((x + 1)(x + 3))}\).