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

Question

Simplify the expression

6 x^{6} - x^{4} + 3 x^{3} - 2 x^{2}

Evaluate

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

Remove the parentheses

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

Multiply

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Multiply the terms

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

x^{6} \times 3 \times 2

Multiply the terms

x^{6} \times 6

Use the commutative property to reorder the terms

6 x^{6}

6 x^{6} - x \times x \times 1 \times (x - 2) (x - 1)

Multiply the terms

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Multiply the terms

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

Rewrite the expression

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

Multiply the terms

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

6 x^{6} - x^{2} (x - 2) (x - 1)

Solution

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Calculate

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

Simplify

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Evaluate

- x^{2} (x - 2)

Apply the distributive property

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

Multiply the terms

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

Use the commutative property to reorder the terms

- x^{3} - (- 2 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

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

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

Apply the distributive property

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

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}

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

Any expression multiplied by 1 remains the same

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

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}

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

Any expression multiplied by 1 remains the same

- x^{4} - (- x^{3}) + 2 x^{3} - 2 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

- x^{4} + x^{3} + 2 x^{3} - 2 x^{2}

Add the terms

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Evaluate

x^{3} + 2 x^{3}

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

(1 + 2) x^{3}

Add the numbers

3 x^{3}

- x^{4} + 3 x^{3} - 2 x^{2}

6 x^{6} - x^{4} + 3 x^{3} - 2 x^{2}

Show Solution

Factor the expression

x^{2} (x + 1) (6 x^{3} - 6 x^{2} + 5 x - 2)

Evaluate

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

Remove the parentheses

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

Multiply

More Steps

Multiply the terms

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

x^{6} \times 3 \times 2

Multiply the terms

x^{6} \times 6

Use the commutative property to reorder the terms

6 x^{6}

6 x^{6} - x \times x \times 1 \times (x - 2) (x - 1)

Any expression multiplied by 1 remains the same

6 x^{6} - x \times x (x - 2) (x - 1)

Multiply the terms

6 x^{6} - x^{2} (x - 2) (x - 1)

Rewrite the expression

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

Factor out x^{2} from the expression

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

Factor the expression

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Evaluate

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

Simplify

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Evaluate

(- x + 2) (x - 1)

Apply the distributive property

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

Multiply the terms

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

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

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

Multiply the terms

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

6 x^{4} - x^{2} + x + 2 x - 2

Add the terms

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Evaluate

x + 2 x

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

(1 + 2) x

Add the numbers

3 x

6 x^{4} - x^{2} + 3 x - 2

Calculate

6 x^{4} - 6 x^{3} + 5 x^{2} - 2 x + 6 x^{3} - 6 x^{2} + 5 x - 2

Rewrite the expression

x \times 6 x^{3} - x \times 6 x^{2} + x \times 5 x - x \times 2 + 6 x^{3} - 6 x^{2} + 5 x - 2

Factor out x from the expression

x (6 x^{3} - 6 x^{2} + 5 x - 2) + 6 x^{3} - 6 x^{2} + 5 x - 2

Factor out 6 x^{3} - 6 x^{2} + 5 x - 2 from the expression

(x + 1) (6 x^{3} - 6 x^{2} + 5 x - 2)

(x + 1) (6 x^{3} - 6 x^{2} + 5 x - 2) x^{2}

Solution

x^{2} (x + 1) (6 x^{3} - 6 x^{2} + 5 x - 2)

Show Solution

Find the roots

x_{1} = - 1, x_{2} = 0, x_{3} \approx 0.567057

Evaluate

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

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

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

Any expression multiplied by 1 remains the same

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

Multiply

More Steps

Multiply the terms

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

x^{6} \times 3 \times 2

Multiply the terms

x^{6} \times 6

Use the commutative property to reorder the terms

6 x^{6}

6 x^{6} - x \times x (x - 2) (x - 1) = 0

Multiply the terms

6 x^{6} - x^{2} (x - 2) (x - 1) = 0

Calculate

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Calculate

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

Simplify

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Evaluate

- x^{2} (x - 2)

Apply the distributive property

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

Multiply the terms

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

Use the commutative property to reorder the terms

- x^{3} - (- 2 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

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

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

Apply the distributive property

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

Multiply the terms

More Steps

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}

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

Any expression multiplied by 1 remains the same

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

Multiply the terms

More Steps

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}

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

Any expression multiplied by 1 remains the same

- x^{4} - (- x^{3}) + 2 x^{3} - 2 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

- x^{4} + x^{3} + 2 x^{3} - 2 x^{2}

Add the terms

More Steps

Evaluate

x^{3} + 2 x^{3}

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

(1 + 2) x^{3}

Add the numbers

3 x^{3}

- x^{4} + 3 x^{3} - 2 x^{2}

6 x^{6} - x^{4} + 3 x^{3} - 2 x^{2} = 0

Factor the expression

x^{2} (x + 1) (6 x^{3} - 6 x^{2} + 5 x - 2) = 0

Separate the equation into 3 possible cases

x^{2} = 0 x + 1 = 0 6 x^{3} - 6 x^{2} + 5 x - 2 = 0

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

x = 0 x + 1 = 0 6 x^{3} - 6 x^{2} + 5 x - 2 = 0

Solve the equation

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Evaluate

x + 1 = 0

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

x = 0 - 1

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

x = - 1

x = 0 x = - 1 6 x^{3} - 6 x^{2} + 5 x - 2 = 0

Solve the equation

x = 0 x = - 1 x \approx 0.567057

Solution

x_{1} = - 1, x_{2} = 0, x_{3} \approx 0.567057

Show Solution