Solve (x + 1) ^ 3 + (x - 1) ^ 3

Question

Simplify the expression

2 x^{3} + 6 x

Evaluate

(x + 1)^{3} + (x - 1)^{3}

Expand the expression

x^{3} + 3 x^{2} + 3 x + 1 + (x - 1)^{3}

Expand the expression

x^{3} + 3 x^{2} + 3 x + 1 + x^{3} - 3 x^{2} + 3 x - 1

Add the terms

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Evaluate

x^{3} + x^{3}

Collect like terms by calculating the sum or difference of their coefficients

(1 + 1) x^{3}

Add the numbers

2 x^{3}

2 x^{3} + 3 x^{2} + 3 x + 1 - 3 x^{2} + 3 x - 1

The sum of two opposites equals 0

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Evaluate

3 x^{2} - 3 x^{2}

Collect like terms

(3 - 3) x^{2}

Add the coefficients

0 \times x^{2}

Calculate

0

2 x^{3} + 0 + 3 x + 1 + 3 x - 1

Remove 0

2 x^{3} + 3 x + 1 + 3 x - 1

Add the terms

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Evaluate

3 x + 3 x

Collect like terms by calculating the sum or difference of their coefficients

(3 + 3) x

Add the numbers

6 x

2 x^{3} + 6 x + 1 - 1

Solution

2 x^{3} + 6 x

Show Solution

Factor the expression

2 x (x^{2} + 3)

Evaluate

(x + 1)^{3} + (x - 1)^{3}

Use a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2}) to factor the expression

(x + 1 + x - 1) ((x + 1)^{2} - (x + 1) (x - 1) + (x - 1)^{2})

Evaluate

(x + 1 + x - 1) ((x + 1)^{2} + (- x - 1) (x - 1) + (x - 1)^{2})

Calculate

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Simplify

x + 1 + x - 1

Add the terms

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Evaluate

x + x

Collect like terms by calculating the sum or difference of their coefficients

(1 + 1) x

Add the numbers

2 x

2 x + 1 - 1

Since two opposites add up to 0,remove them form the expression

2 x

2 x ((x + 1)^{2} + (- x - 1) (x - 1) + (x - 1)^{2})

Solution

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Simplify

(x + 1)^{2} + (- x - 1) (x - 1) + (x - 1)^{2}

Simplify

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Simplify

(- x - 1) (x - 1)

Apply the distributive property

- x \times x - x (- 1) - x - (- 1)

Multiply the terms

- x^{2} - x (- 1) - x - (- 1)

Multiplying or dividing an even number of negative terms equals a positive

- x^{2} + x - x - (- 1)

When there is - in front of an expression in parentheses change the sign of each term of the expression and remove the parentheses

- x^{2} + x - x + 1

(x + 1)^{2} - x^{2} + x - x + 1 + (x - 1)^{2}

The sum of two opposites equals 0

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Evaluate

x - x

Collect like terms

(1 - 1) x

Add the coefficients

0 \times x

Calculate

0

(x + 1)^{2} - x^{2} + 0 + 1 + (x - 1)^{2}

Remove 0

(x + 1)^{2} - x^{2} + 1 + (x - 1)^{2}

Expand the expression

x^{2} + 2 x + 1 - x^{2} + 1 + (x - 1)^{2}

Expand the expression

x^{2} + 2 x + 1 - x^{2} + 1 + x^{2} - 2 x + 1

Calculate the sum or difference

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Evaluate

x^{2} - x^{2} + x^{2}

Collect like terms by calculating the sum or difference of their coefficients

(1 - 1 + 1) x^{2}

Calculate the sum or difference

x^{2}

x^{2} + 2 x + 1 + 1 - 2 x + 1

The sum of two opposites equals 0

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Evaluate

2 x - 2 x

Collect like terms

(2 - 2) x

Add the coefficients

0 \times x

Calculate

0

x^{2} + 0 + 1 + 1 + 1

Remove 0

x^{2} + 1 + 1 + 1

Add the numbers

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Evaluate

1 + 1 + 1

Add the numbers

2 + 1

Add the numbers

3

x^{2} + 3

2 x (x^{2} + 3)

Show Solution

Find the roots

x_{1} = - 3 \times i, x_{2} = 3 \times i, x_{3} = 0

Alternative Form

x_{1} \approx - 1.732051 i, x_{2} \approx 1.732051 i, x_{3} = 0

Evaluate

(x + 1)^{3} + (x - 1)^{3}

To find the roots of the expression,set the expression equal to 0

(x + 1)^{3} + (x - 1)^{3} = 0

Factor the expression

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Evaluate

(x + 1)^{3} + (x - 1)^{3}

Use a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2}) to factor the expression

(x + 1 + x - 1) ((x + 1)^{2} - (x + 1) (x - 1) + (x - 1)^{2})

Evaluate

(x + 1 + x - 1) ((x + 1)^{2} + (- x - 1) (x - 1) + (x - 1)^{2})

(x + 1 + x - 1) ((x + 1)^{2} + (- x - 1) (x - 1) + (x - 1)^{2}) = 0

Separate the equation into 2 possible cases

x + 1 + x - 1 = 0 (x + 1)^{2} + (- x - 1) (x - 1) + (x - 1)^{2} = 0

Solve the equation

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Evaluate

x + 1 + x - 1 = 0

Calculate the sum or difference

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Evaluate

x + 1 + x - 1

Add the terms

2 x + 1 - 1

Since two opposites add up to 0,remove them form the expression

2 x

2 x = 0

Rewrite the expression

x = 0

x = 0 (x + 1)^{2} + (- x - 1) (x - 1) + (x - 1)^{2} = 0

Solve the equation

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Evaluate

(x + 1)^{2} + (- x - 1) (x - 1) + (x - 1)^{2} = 0

Calculate

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Evaluate

(x + 1)^{2} + (- x - 1) (x - 1) + (x - 1)^{2}

Expand the expression

x^{2} + 2 x + 1 + (- x - 1) (x - 1) + (x - 1)^{2}

Expand the expression

x^{2} + 2 x + 1 + 1 - x^{2} + (x - 1)^{2}

Expand the expression

x^{2} + 2 x + 1 + 1 - x^{2} + x^{2} - 2 x + 1

Calculate the sum or difference

x^{2} + 2 x + 1 + 1 - 2 x + 1

The sum of two opposites equals 0

x^{2} + 0 + 1 + 1 + 1

Remove 0

x^{2} + 1 + 1 + 1

Add the numbers

x^{2} + 3

x^{2} + 3 = 0

Move the constant to the right-hand side and change its sign

x^{2} = 0 - 3

Removing 0 doesn't change the value,so remove it from the expression

x^{2} = - 3

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm - 3

Simplify the expression

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Evaluate

- 3

Evaluate the power

3 \times - 1

Evaluate the power

3 \times i

x = \pm (3 \times i)

Separate the equation into 2 possible cases

x = 3 \times i x = - 3 \times i

x = 0 x = 3 \times i x = - 3 \times i

Solution

x_{1} = - 3 \times i, x_{2} = 3 \times i, x_{3} = 0

Alternative Form

x_{1} \approx - 1.732051 i, x_{2} \approx 1.732051 i, x_{3} = 0

Show Solution