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

Question

Simplify the expression

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

Evaluate

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

Multiply the first two terms

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

Multiply the terms

More Steps

Evaluate

x^{3} (1 - 2 x)

Apply the distributive property

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

Any expression multiplied by 1 remains the same

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

Multiply the terms

More Steps

Evaluate

x^{3} \times 2 x

Use the commutative property to reorder the terms

2 x^{3} \times x

Multiply the terms

2 x^{4}

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

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

Apply the distributive property

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

Any expression multiplied by 1 remains the same

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

Multiply the terms

More Steps

Evaluate

x^{4} \times x

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

x^{4 + 1}

Add the numbers

x^{5}

x^{4} - x^{3} - 2 x^{5} - (- 2 x^{4} \times 1)

Any expression multiplied by 1 remains the same

x^{4} - x^{3} - 2 x^{5} - (- 2 x^{4})

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^{5} + 2 x^{4}

Solution

More Steps

Evaluate

x^{4} + 2 x^{4}

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

(1 + 2) x^{4}

Add the numbers

3 x^{4}

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

Show Solution

Find the roots

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

Alternative Form

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

Evaluate

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

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

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

Calculate

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

Multiply the first two terms

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

Separate the equation into 3 possible cases

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

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

x = 0 1 - 2 x = 0 x - 1 = 0

Solve the equation

More Steps

Evaluate

1 - 2 x = 0

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

- 2 x = 0 - 1

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

- 2 x = - 1

Change the signs on both sides of the equation

2 x = 1

Divide both sides

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

Divide the numbers

x = \frac{1}{2}

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

Solve the equation

More Steps

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

Solution

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

Alternative Form

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

Show Solution