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

Since <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> has no factors besides <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The LCM of <math><mstyle displaystyle="true"><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</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><mn>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mn>1</mn></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><mn>1</mn><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.

The Least Common Multiple <math><mstyle displaystyle="true"><mi>LCM</mi></mstyle></math> of some numbers is the smallest number that the numbers are factors of.

Multiply each term in <math><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><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.

Rewrite using the commutative property of multiplication.

Combine <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math> .

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

Cancel the common factor.

Rewrite the expression.

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

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

Use the power rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mstyle></math> to combine exponents.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Rewrite using the commutative property of multiplication.

Multiply <math><mstyle displaystyle="true"><mn>2</mn><mfrac><mrow><mn>4</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

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><mn>1</mn><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.

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mn>8</mn></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"><mn>2</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><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><mn>1</mn><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 and combine like terms.

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

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

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

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</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>5</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> from both sides of the equation.

Simplify each term.

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> .

Expand <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><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 and combine like terms.

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> .

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

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

Apply the distributive property.

Simplify.

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

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

Subtract <math><mstyle displaystyle="true"><mn>5</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>4</mn><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>8</mn><mi>x</mi></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> to both sides of the equation.

Add <math><mstyle displaystyle="true"><mn>10</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> .

Factor the left side of the equation.

Factor <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>15</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>4</mn><mi>x</mi></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>15</mn></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn><mrow><mo>(</mo><mo>-</mo><mn>15</mn><mo>)</mo></mrow></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><mrow><mo>(</mo><mn>3</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mo>-</mo><mrow><mo>(</mo><mo>-</mo><mn>4</mn><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><mrow><mo>(</mo><mn>3</mn><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>4</mn><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>1</mn><mrow><mo>(</mo><mo>-</mo><mn>15</mn><mo>)</mo></mrow></mstyle></math> .

Factor.

Factor by grouping.

For a polynomial of the form <math><mstyle displaystyle="true"><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mstyle></math> , rewrite the middle term as a sum of two terms whose product is <math><mstyle displaystyle="true"><mi>a</mi><mo>⋅</mo><mi>c</mi><mo>=</mo><mn>3</mn><mo>⋅</mo><mo>-</mo><mn>15</mn><mo>=</mo><mo>-</mo><mn>45</mn></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mo>-</mo><mn>4</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mo>-</mo><mn>4</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><mn>4</mn><mi>x</mi></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mo>-</mo><mn>4</mn></mstyle></math> as <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> plus <math><mstyle displaystyle="true"><mo>-</mo><mn>9</mn></mstyle></math>

Apply the distributive property.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

Factor out the greatest common factor (GCF) from each group.

Factor the polynomial by factoring out the greatest common factor, <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mstyle></math> .

Remove unnecessary parentheses.

If any individual factor on the left side of the equation is equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , the entire expression will be equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Set <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and solve for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Set <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Solve <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></mstyle></math> for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> from both sides of the equation.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>=</mo><mo>-</mo><mn>5</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>=</mo><mo>-</mo><mn>5</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Simplify the left side.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>3</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.

Move the negative in front of the fraction.

Set <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mn>3</mn></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and solve for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Set <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mn>3</mn></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> to both sides of the equation.

The final solution is all the values that make <math><mstyle displaystyle="true"><mo>-</mo><mrow><mo>(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mstyle></math> true.

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

Mixed Number Form:

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