Solve (3x 1)(x 1)(x-2)(3x-2)

Question

Simplify the expression

9 x^{4} - 24 x^{3} + 12 x^{2}

Evaluate

(3 x \times 1) (x \times 1) (x - 2) (3 x - 2)

Remove the parentheses

3 x \times 1 \times x \times 1 \times (x - 2) (3 x - 2)

Rewrite the expression

3 x \times x (x - 2) (3 x - 2)

Multiply the terms

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

Multiply the terms

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Evaluate

3 x^{2} (x - 2)

Apply the distributive property

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

Multiply the terms

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Evaluate

x^{2} \times x

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

x^{2 + 1}

Add the numbers

x^{3}

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

Multiply the numbers

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

(3 x^{3} - 6 x^{2}) (3 x - 2)

Apply the distributive property

3 x^{3} \times 3 x - 3 x^{3} \times 2 - 6 x^{2} \times 3 x - (- 6 x^{2} \times 2)

Multiply the terms

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Evaluate

3 x^{3} \times 3 x

Multiply the numbers

9 x^{3} \times x

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}

9 x^{4}

9 x^{4} - 3 x^{3} \times 2 - 6 x^{2} \times 3 x - (- 6 x^{2} \times 2)

Multiply the numbers

9 x^{4} - 6 x^{3} - 6 x^{2} \times 3 x - (- 6 x^{2} \times 2)

Multiply the terms

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Evaluate

- 6 x^{2} \times 3 x

Multiply the numbers

- 18 x^{2} \times x

Multiply the terms

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Evaluate

x^{2} \times x

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

x^{2 + 1}

Add the numbers

x^{3}

- 18 x^{3}

9 x^{4} - 6 x^{3} - 18 x^{3} - (- 6 x^{2} \times 2)

Multiply the numbers

9 x^{4} - 6 x^{3} - 18 x^{3} - (- 12 x^{2})

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

9 x^{4} - 6 x^{3} - 18 x^{3} + 12 x^{2}

Solution

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Evaluate

- 6 x^{3} - 18 x^{3}

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

(- 6 - 18) x^{3}

Subtract the numbers

- 24 x^{3}

9 x^{4} - 24 x^{3} + 12 x^{2}

Show Solution

Find the roots

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

Alternative Form

x_{1} = 0, x_{2} = 0. \dot{6}, x_{3} = 2

Evaluate

(3 x \times 1) (x \times 1) (x - 2) (3 x - 2)

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

(3 x \times 1) (x \times 1) (x - 2) (3 x - 2) = 0

Multiply the terms

3 x (x \times 1) (x - 2) (3 x - 2) = 0

Any expression multiplied by 1 remains the same

3 x \times x (x - 2) (3 x - 2) = 0

Multiply the terms

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

Elimination the left coefficient

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

Separate the equation into 3 possible cases

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

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

x = 0 x - 2 = 0 3 x - 2 = 0

Solve the equation

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Evaluate

x - 2 = 0

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

x = 0 + 2

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

x = 2

x = 0 x = 2 3 x - 2 = 0

Solve the equation

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Evaluate

3 x - 2 = 0

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

3 x = 0 + 2

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

3 x = 2

Divide both sides

\frac{3 x}{3} = \frac{2}{3}

Divide the numbers

x = \frac{2}{3}

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

Solution

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

Alternative Form

x_{1} = 0, x_{2} = 0. \dot{6}, x_{3} = 2

Show Solution