Solve (x-3)(x^2 3x 9)*(4x^2-1)

Question

Simplify the expression

108 x^{6} - 27 x^{4} - 324 x^{5} + 81 x^{3}

Evaluate

(x - 3) (x^{2} \times 3 x \times 9) (4 x^{2} - 1)

Remove the parentheses

(x - 3) x^{2} \times 3 x \times 9 (4 x^{2} - 1)

Multiply the terms with the same base by adding their exponents

(x - 3) x^{2 + 1} \times 3 \times 9 (4 x^{2} - 1)

Add the numbers

(x - 3) x^{3} \times 3 \times 9 (4 x^{2} - 1)

Multiply the terms

(x - 3) x^{3} \times 27 (4 x^{2} - 1)

Use the commutative property to reorder the terms

(x - 3) \times 27 x^{3} (4 x^{2} - 1)

Multiply the first two terms

27 x^{3} (x - 3) (4 x^{2} - 1)

Multiply the terms

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Evaluate

27 x^{3} (x - 3)

Apply the distributive property

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

Multiply the terms

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Evaluate

x^{3} \times x

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

x^{3 + 1}

Add the numbers

x^{4}

27 x^{4} - 27 x^{3} \times 3

Multiply the numbers

27 x^{4} - 81 x^{3}

(27 x^{4} - 81 x^{3}) (4 x^{2} - 1)

Apply the distributive property

27 x^{4} \times 4 x^{2} - 27 x^{4} \times 1 - 81 x^{3} \times 4 x^{2} - (- 81 x^{3} \times 1)

Multiply the terms

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Evaluate

27 x^{4} \times 4 x^{2}

Multiply the numbers

108 x^{4} \times x^{2}

Multiply the terms

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Evaluate

x^{4} \times x^{2}

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

x^{4 + 2}

Add the numbers

x^{6}

108 x^{6}

108 x^{6} - 27 x^{4} \times 1 - 81 x^{3} \times 4 x^{2} - (- 81 x^{3} \times 1)

Any expression multiplied by 1 remains the same

108 x^{6} - 27 x^{4} - 81 x^{3} \times 4 x^{2} - (- 81 x^{3} \times 1)

Multiply the terms

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Evaluate

- 81 x^{3} \times 4 x^{2}

Multiply the numbers

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

Multiply the terms

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Evaluate

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

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

x^{3 + 2}

Add the numbers

x^{5}

- 324 x^{5}

108 x^{6} - 27 x^{4} - 324 x^{5} - (- 81 x^{3} \times 1)

Any expression multiplied by 1 remains the same

108 x^{6} - 27 x^{4} - 324 x^{5} - (- 81 x^{3})

Solution

108 x^{6} - 27 x^{4} - 324 x^{5} + 81 x^{3}

Show Solution

Factor the expression

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

Evaluate

(x - 3) (x^{2} \times 3 x \times 9) (4 x^{2} - 1)

Remove the parentheses

(x - 3) x^{2} \times 3 x \times 9 (4 x^{2} - 1)

Multiply

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Multiply the terms

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

x^{3} \times 3 \times 9

Multiply the terms

x^{3} \times 27

Use the commutative property to reorder the terms

27 x^{3}

(x - 3) \times 27 x^{3} (4 x^{2} - 1)

Multiply the first two terms

27 x^{3} (x - 3) (4 x^{2} - 1)

Solution

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

Show Solution

Find the roots

x_{1} = - \frac{1}{2}, x_{2} = 0, x_{3} = \frac{1}{2}, x_{4} = 3

Alternative Form

x_{1} = - 0.5, x_{2} = 0, x_{3} = 0.5, x_{4} = 3

Evaluate

(x - 3) (x^{2} \times 3 x \times 9) (4 x^{2} - 1)

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

(x - 3) (x^{2} \times 3 x \times 9) (4 x^{2} - 1) = 0

Multiply

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Multiply the terms

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

x^{3} \times 3 \times 9

Multiply the terms

x^{3} \times 27

Use the commutative property to reorder the terms

27 x^{3}

(x - 3) \times 27 x^{3} (4 x^{2} - 1) = 0

Multiply the first two terms

27 x^{3} (x - 3) (4 x^{2} - 1) = 0

Elimination the left coefficient

x^{3} (x - 3) (4 x^{2} - 1) = 0

Separate the equation into 3 possible cases

x^{3} = 0 x - 3 = 0 4 x^{2} - 1 = 0

The only way a power can be 0 is when the base equals 0

x = 0 x - 3 = 0 4 x^{2} - 1 = 0

Solve the equation

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Evaluate

x - 3 = 0

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

x = 0 + 3

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

x = 3

x = 0 x = 3 4 x^{2} - 1 = 0

Solve the equation

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Evaluate

4 x^{2} - 1 = 0

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

4 x^{2} = 0 + 1

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

4 x^{2} = 1

Divide both sides

\frac{4 x ^{2}}{4} = \frac{1}{4}

Divide the numbers

x^{2} = \frac{1}{4}

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

x = \pm \frac{1}{4}

Simplify the expression

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Evaluate

\frac{1}{4}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{1}{4}

Simplify the radical expression

\frac{1}{4}

Simplify the radical expression

\frac{1}{2}

x = \pm \frac{1}{2}

Separate the equation into 2 possible cases

x = \frac{1}{2} x = - \frac{1}{2}

x = 0 x = 3 x = \frac{1}{2} x = - \frac{1}{2}

Solution

x_{1} = - \frac{1}{2}, x_{2} = 0, x_{3} = \frac{1}{2}, x_{4} = 3

Alternative Form

x_{1} = - 0.5, x_{2} = 0, x_{3} = 0.5, x_{4} = 3

Show Solution