Solve -x^3 3x^2 9x-27

Question

Simplify the expression

- 27 x^{6} - 27

Evaluate

- x^{3} \times 3 x^{2} \times 9 x - 27

Solution

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Evaluate

- x^{3} \times 3 x^{2} \times 9 x

Multiply the terms with the same base by adding their exponents

- x^{3 + 2 + 1} \times 3 \times 9

Add the numbers

- x^{6} \times 3 \times 9

Multiply the terms

- x^{6} \times 27

Use the commutative property to reorder the terms

- 27 x^{6}

- 27 x^{6} - 27

Show Solution

Factor the expression

- 27 (x^{2} + 1) (x^{4} - x^{2} + 1)

Evaluate

- x^{3} \times 3 x^{2} \times 9 x - 27

Evaluate

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Evaluate

x^{3} \times 3 x^{2} \times 9 x

Multiply the terms with the same base by adding their exponents

x^{3 + 2 + 1} \times 3 \times 9

Add the numbers

x^{6} \times 3 \times 9

Multiply the terms

x^{6} \times 27

Use the commutative property to reorder the terms

27 x^{6}

- 27 x^{6} - 27

Factor out - 27 from the expression

- 27 (x^{6} + 1)

Solution

More Steps

Evaluate

x^{6} + 1

Rewrite the expression in exponential form

x^{6} + 1^{3}

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

(x^{2} + 1) (x^{4} - x^{2} \times 1 + 1^{2})

Any expression multiplied by 1 remains the same

(x^{2} + 1) (x^{4} - x^{2} + 1^{2})

1 raised to any power equals to 1

(x^{2} + 1) (x^{4} - x^{2} + 1)

- 27 (x^{2} + 1) (x^{4} - x^{2} + 1)

Show Solution

Find the roots

x_{1} = - \frac{3}{2} - \frac{1}{2} i, x_{2} = \frac{3}{2} + \frac{1}{2} i

Alternative Form

x_{1} \approx - 0.866025 - 0.5 i, x_{2} \approx 0.866025 + 0.5 i

Evaluate

- x^{3} \times 3 x^{2} \times 9 x - 27

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

- x^{3} \times 3 x^{2} \times 9 x - 27 = 0

Multiply

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Multiply the terms

x^{3} \times 3 x^{2} \times 9 x

Multiply the terms with the same base by adding their exponents

x^{3 + 2 + 1} \times 3 \times 9

Add the numbers

x^{6} \times 3 \times 9

Multiply the terms

x^{6} \times 27

Use the commutative property to reorder the terms

27 x^{6}

- 27 x^{6} - 27 = 0

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

- 27 x^{6} = 0 + 27

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

- 27 x^{6} = 27

Change the signs on both sides of the equation

27 x^{6} = - 27

Divide both sides

\frac{27 x ^{6}}{27} = \frac{- 27}{27}

Divide the numbers

x^{6} = \frac{- 27}{27}

Divide the numbers

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Evaluate

\frac{- 27}{27}

Reduce the numbers

\frac{- 1}{1}

Calculate

- 1

x^{6} = - 1

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

x = \pm 6 - 1

Simplify the expression

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Evaluate

6 - 1

Rewrite the expression

1 \times (\frac{3}{2} + \frac{1}{2} i)

Any expression multiplied by 1 remains the same

\frac{3}{2} + \frac{1}{2} i

x = \pm (\frac{3}{2} + \frac{1}{2} i)

Separate the equation into 2 possible cases

x = \frac{3}{2} + \frac{1}{2} i x = - \frac{3}{2} - \frac{1}{2} i

Solution

x_{1} = - \frac{3}{2} - \frac{1}{2} i, x_{2} = \frac{3}{2} + \frac{1}{2} i

Alternative Form

x_{1} \approx - 0.866025 - 0.5 i, x_{2} \approx 0.866025 + 0.5 i

Show Solution