Add and Subtract Rational Expressions

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Key Ideas Easy Medium Hard

Rewrite each fraction over the LCD.

Subtracting a fraction means distribute the minus.

What to Remember

One denominator

Factor each denominator.

The LCD uses the highest power of every factor.

Rewrite, then add

Rewrite each fraction over the LCD.

Add or subtract the numerators. The denominator stays.

A minus sign

Put parentheses around the numerator you subtract.

Distribute the minus before you combine like terms.

Cancel at the end

Factor the new numerator and cancel if you can.

A canceled value is still excluded.

Level 1

Easy

Different linear denominators

Example

Simplify. xx + 2 + 3x − 1

The LCD is (x + 2)(x − 1).

Rewrite each fraction over the LCD.

x² − x(x + 2)(x − 1)

3x + 6(x + 2)(x − 1)

Combine the numerators.

x² + 2x + 6(x + 2)(x − 1)

x ≠ −2 and x ≠ 1

Answer: x² + 2x + 6(x + 2)(x − 1), x ≠ −2 and x ≠ 1

Simplify. 2x + 1 + 4x − 3

Show Answer

The LCD is (x + 1)(x − 3).

Rewrite each fraction over the LCD.

2x − 6(x + 1)(x − 3)

4x + 4(x + 1)(x − 3)

Combine the numerators.

6x − 2(x + 1)(x − 3)

Factor the numerator.

2(3x − 1)(x + 1)(x − 3)

x ≠ −1 and x ≠ 3

Answer: 2(3x − 1)(x + 1)(x − 3), x ≠ −1 and x ≠ 3

Simplify. 5x − 2 + 1x + 4

Show Answer

The LCD is (x − 2)(x + 4).

Rewrite each fraction over the LCD.

5x + 20(x − 2)(x + 4)

x − 2(x − 2)(x + 4)

Combine the numerators.

6x + 18(x − 2)(x + 4)

Factor the numerator.

6(x + 3)(x − 2)(x + 4)

x ≠ −4 and x ≠ 2

Answer: 6(x + 3)(x − 2)(x + 4), x ≠ −4 and x ≠ 2

Simplify. xx + 5 − (2x − 1)

Show Answer

The LCD is (x + 5)(x − 1).

Rewrite each fraction over the LCD.

x² − x(x + 5)(x − 1)

−(2x + 10)(x + 5)(x − 1)

Distribute the minus.

−2x − 10(x + 5)(x − 1)

Combine the numerators.

x² − 3x − 10(x + 5)(x − 1)

Factor the numerator.

(x − 5)(x + 2)(x + 5)(x − 1)

x ≠ −5 and x ≠ 1

Answer: (x − 5)(x + 2)(x + 5)(x − 1), x ≠ −5 and x ≠ 1

Simplify. 3x + 2 + 2x + 6

Show Answer

The LCD is (x + 2)(x + 6).

Rewrite each fraction over the LCD.

3x + 18(x + 2)(x + 6)

2x + 4(x + 2)(x + 6)

Combine the numerators.

5x + 22(x + 2)(x + 6)

Factor the numerator.

(5x + 22)(x + 2)(x + 6)

x ≠ −6 and x ≠ −2

Answer: (5x + 22)(x + 2)(x + 6), x ≠ −6 and x ≠ −2

Simplify. 4x − 5 − (1x + 1)

Show Answer

The LCD is (x − 5)(x + 1).

Rewrite each fraction over the LCD.

4x + 4(x − 5)(x + 1)

−(x − 5)(x − 5)(x + 1)

Distribute the minus.

5 − x(x − 5)(x + 1)

Combine the numerators.

3x + 9(x − 5)(x + 1)

Factor the numerator.

3(x + 3)(x − 5)(x + 1)

x ≠ −1 and x ≠ 5

Answer: 3(x + 3)(x − 5)(x + 1), x ≠ −1 and x ≠ 5

Level 2

Medium

A denominator that factors

Example

Simplify. 4xx² − 1 − (2x − 1)

Factor x² − 1.

(x − 1)(x + 1)

The LCD is (x − 1)(x + 1).

Rewrite each fraction over the LCD.

4x(x − 1)(x + 1)

−(2x + 2)(x − 1)(x + 1)

Distribute the minus.

−2x − 2(x − 1)(x + 1)

Combine the numerators.

2x − 2(x − 1)(x + 1)

Factor the numerator.

2(x − 1)(x − 1)(x + 1)

Cancel the common factor. The canceled value is still excluded.

2(x + 1)

x ≠ −1 and x ≠ 1

Answer: 2(x + 1), x ≠ −1 and x ≠ 1

Simplify. 2x − 1 − (x + 3x² − 1)

Show Answer

Factor x² − 1.

(x − 1)(x + 1)

The LCD is (x − 1)(x + 1).

Rewrite each fraction over the LCD.

2x + 2(x − 1)(x + 1)

−(x + 3)(x − 1)(x + 1)

Distribute the minus.

−x − 3(x − 1)(x + 1)

Combine the numerators.

x − 1(x − 1)(x + 1)

Factor the numerator.

(x − 1)(x − 1)(x + 1)

Cancel the common factor. The canceled value is still excluded.

1(x + 1)

x ≠ −1 and x ≠ 1

Answer: 1(x + 1), x ≠ −1 and x ≠ 1

Simplify. 2xx² − 9 − (1x − 3)

Show Answer

Factor x² − 9.

(x − 3)(x + 3)

The LCD is (x − 3)(x + 3).

Rewrite each fraction over the LCD.

2x(x − 3)(x + 3)

−(x + 3)(x − 3)(x + 3)

Distribute the minus.

−x − 3(x − 3)(x + 3)

Combine the numerators.

x − 3(x − 3)(x + 3)

Factor the numerator.

(x − 3)(x − 3)(x + 3)

Cancel the common factor. The canceled value is still excluded.

1(x + 3)

x ≠ −3 and x ≠ 3

Answer: 1(x + 3), x ≠ −3 and x ≠ 3

Simplify. x + 2x² − 16 + 1x + 4

Show Answer

Factor x² − 16.

(x − 4)(x + 4)

The LCD is (x − 4)(x + 4).

Rewrite each fraction over the LCD.

x + 2(x − 4)(x + 4)

x − 4(x − 4)(x + 4)

Combine the numerators.

2x − 2(x − 4)(x + 4)

Factor the numerator.

2(x − 1)(x − 4)(x + 4)

x ≠ −4 and x ≠ 4

Answer: 2(x − 1)(x − 4)(x + 4), x ≠ −4 and x ≠ 4

Simplify. 5x + 1 − (2xx² + 3x + 2)

Show Answer

Factor x² + 3x + 2.

(x + 1)(x + 2)

The LCD is (x + 1)(x + 2).

Rewrite each fraction over the LCD.

5x + 10(x + 1)(x + 2)

−(2x)(x + 1)(x + 2)

Distribute the minus.

−2x(x + 1)(x + 2)

Combine the numerators.

3x + 10(x + 1)(x + 2)

Factor the numerator.

(3x + 10)(x + 1)(x + 2)

x ≠ −2 and x ≠ −1

Answer: (3x + 10)(x + 1)(x + 2), x ≠ −2 and x ≠ −1

Simplify. 3xx² + 5x + 6 + 2x + 2

Show Answer

Factor x² + 5x + 6.

(x + 2)(x + 3)

The LCD is (x + 2)(x + 3).

Rewrite each fraction over the LCD.

3x(x + 2)(x + 3)

2x + 6(x + 2)(x + 3)

Combine the numerators.

5x + 6(x + 2)(x + 3)

Factor the numerator.

(5x + 6)(x + 2)(x + 3)

x ≠ −3 and x ≠ −2

Answer: (5x + 6)(x + 2)(x + 3), x ≠ −3 and x ≠ −2

Level 3

Hard

Both denominators factor

Example

Simplify. xx² − 9 − (1x² + 6x + 9)

Factor x² − 9.

(x − 3)(x + 3)

Factor x² + 6x + 9.

(x + 3)²

The LCD is (x − 3)(x + 3)².

Rewrite each fraction over the LCD.

x² + 3x(x − 3)(x + 3)²

−(x − 3)(x − 3)(x + 3)²

Distribute the minus.

3 − x(x − 3)(x + 3)²

Combine the numerators.

x² + 2x + 3(x − 3)(x + 3)²

x ≠ −3 and x ≠ 3

Answer: x² + 2x + 3(x − 3)(x + 3)², x ≠ −3 and x ≠ 3

Simplify. 2x² − 4 − (x + 1x² − 4x + 4)

Show Answer

Factor x² − 4.

(x − 2)(x + 2)

Factor x² − 4x + 4.

(x − 2)²

The LCD is (x − 2)²(x + 2).

Rewrite each fraction over the LCD.

2x − 4(x − 2)²(x + 2)

−(x² + 3x + 2)(x − 2)²(x + 2)

Distribute the minus.

−x² − 3x − 2(x − 2)²(x + 2)

Combine the numerators.

−x² − x − 6(x − 2)²(x + 2)

x ≠ −2 and x ≠ 2

Answer: −x² − x − 6(x − 2)²(x + 2), x ≠ −2 and x ≠ 2

Simplify. xx² − 3x − 4 − (3x² + 2x + 1)

Show Answer

Factor x² − 3x − 4.

(x + 1)(x − 4)

Factor x² + 2x + 1.

(x + 1)²

The LCD is (x + 1)²(x − 4).

Rewrite each fraction over the LCD.

x² + x(x + 1)²(x − 4)

−(3x − 12)(x + 1)²(x − 4)

Distribute the minus.

12 − 3x(x + 1)²(x − 4)

Combine the numerators.

x² − 2x + 12(x + 1)²(x − 4)

x ≠ −1 and x ≠ 4

Answer: x² − 2x + 12(x + 1)²(x − 4), x ≠ −1 and x ≠ 4

Simplify. 4x² + 4x − 5 + x + 2x² + 10x + 25

Show Answer

Factor x² + 4x − 5.

(x − 1)(x + 5)

Factor x² + 10x + 25.

(x + 5)²

The LCD is (x − 1)(x + 5)².

Rewrite each fraction over the LCD.

4x + 20(x − 1)(x + 5)²

x² + x − 2(x − 1)(x + 5)²

Combine the numerators.

x² + 5x + 18(x − 1)(x + 5)²

x ≠ −5 and x ≠ 1

Answer: x² + 5x + 18(x − 1)(x + 5)², x ≠ −5 and x ≠ 1

Simplify. x − 2x² − x − 6 − (2x² − 6x + 9)

Show Answer

Factor x² − x − 6.

(x + 2)(x − 3)

Factor x² − 6x + 9.

(x − 3)²

The LCD is (x + 2)(x − 3)².

Rewrite each fraction over the LCD.

x² − 5x + 6(x + 2)(x − 3)²

−(2x + 4)(x + 2)(x − 3)²

Distribute the minus.

−2x − 4(x + 2)(x − 3)²

Combine the numerators.

x² − 7x + 2(x + 2)(x − 3)²

x ≠ −2 and x ≠ 3

Answer: x² − 7x + 2(x + 2)(x − 3)², x ≠ −2 and x ≠ 3

Simplify. 3xx² + 2x − 8 − (1x² + 8x + 16)

Show Answer

Factor x² + 2x − 8.

(x − 2)(x + 4)

Factor x² + 8x + 16.

(x + 4)²

The LCD is (x − 2)(x + 4)².

Rewrite each fraction over the LCD.

3x² + 12x(x − 2)(x + 4)²

−(x − 2)(x − 2)(x + 4)²

Distribute the minus.

2 − x(x − 2)(x + 4)²

Combine the numerators.

3x² + 11x + 2(x − 2)(x + 4)²

x ≠ −4 and x ≠ 2

Answer: 3x² + 11x + 2(x − 2)(x + 4)², x ≠ −4 and x ≠ 2

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