Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of <math><mstyle displaystyle="true"><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn></mstyle></math> is the result of multiplying all prime factors the greatest number of times they occur in either number.

The factor for <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mi>a</mi></mstyle></math> is <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mi>a</mi></mstyle></math> itself.

The factor for <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mi>a</mi></mstyle></math> is <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mi>a</mi></mstyle></math> itself.

The factor for <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mn>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mn>1</mn></mstyle></math> itself.

The LCM of <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo>-</mo><mn>1</mn></mstyle></math> is the result of multiplying all factors the greatest number of times they occur in either term.

Multiply each term in <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mi>a</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> .

Simplify the left side.

Simplify each term.

Cancel the common factor of <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mi>a</mi></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mi>a</mi></mstyle></math> out of <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Expand <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify each term.

Multiply <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Move <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the left of <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><mi>x</mi></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Move <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the left of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><mi>a</mi></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><mi>a</mi></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mi>a</mi></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mi>a</mi></mstyle></math> out of <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Expand <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify each term.

Multiply <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Move <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the left of <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><mi>x</mi></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mi>a</mi><mo>⋅</mo><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify by adding terms.

Combine the opposite terms in <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mi>x</mi><mo>+</mo><mi>a</mi><mi>x</mi><mo>-</mo><mi>a</mi><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mi>x</mi><mo>-</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>a</mi></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mi>a</mi><mi>x</mi></mstyle></math> from <math><mstyle displaystyle="true"><mi>a</mi><mi>x</mi></mstyle></math> .

Add <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mi>a</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> .

Add <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mi>x</mi><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> and <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Simplify the right side.

Cancel the common factor of <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mn>1</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mn>1</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Expand <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>a</mi><mo>)</mo></mrow></mstyle></math> using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify terms.

Combine the opposite terms in <math><mstyle displaystyle="true"><mi>x</mi><mo>⋅</mo><mi>x</mi><mo>+</mo><mi>x</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>x</mi><mo>-</mo><mi>a</mi><mo>⋅</mo><mi>a</mi></mstyle></math> .

Reorder the factors in the terms <math><mstyle displaystyle="true"><mi>x</mi><mi>a</mi></mstyle></math> and <math><mstyle displaystyle="true"><mo>-</mo><mi>a</mi><mi>x</mi></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mi>a</mi><mi>x</mi></mstyle></math> from <math><mstyle displaystyle="true"><mi>a</mi><mi>x</mi></mstyle></math> .

Add <math><mstyle displaystyle="true"><mi>x</mi><mo>⋅</mo><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Simplify each term.

Multiply <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> by <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> by adding the exponents.

Move <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> by <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> .

Simplify by multiplying through.

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Move all terms containing <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> to the left side of the equation.

Subtract <math><mstyle displaystyle="true"><mn>2</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> from both sides of the equation.

Combine the opposite terms in <math><mstyle displaystyle="true"><mn>2</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>2</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> from <math><mstyle displaystyle="true"><mn>2</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Divide each term in <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> .

Simplify the left side.

Cancel the common factor of <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> .

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify the right side.

Cancel the common factor of <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> .

Cancel the common factor.

Divide <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

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