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

Question

Simplify the expression

2 y^{4} - 4 y^{3}

Evaluate

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

Remove the parentheses

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

Multiply the terms

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Multiply the terms

y \times 1 \times y^{2} \times 2

Rewrite the expression

y \times y^{2} \times 2

Multiply the terms with the same base by adding their exponents

y^{1 + 2} \times 2

Add the numbers

y^{3} \times 2

Use the commutative property to reorder the terms

2 y^{3}

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

Expand the expression

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Calculate

2 y^{3} (y - 1)

Apply the distributive property

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

Multiply the terms

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Evaluate

y^{3} \times y

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

y^{3 + 1}

Add the numbers

y^{4}

2 y^{4} - 2 y^{3} \times 1

Any expression multiplied by 1 remains the same

2 y^{4} - 2 y^{3}

2 y^{4} - 2 y^{3} - 2 y^{3}

Solution

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Evaluate

- 2 y^{3} - 2 y^{3}

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

(- 2 - 2) y^{3}

Subtract the numbers

- 4 y^{3}

2 y^{4} - 4 y^{3}

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Factor the expression

2 y^{3} (y - 2)

Evaluate

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

Remove the parentheses

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

Any expression multiplied by 1 remains the same

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

Multiply

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Multiply the terms

y \times y^{2} \times 2

Multiply the terms with the same base by adding their exponents

y^{1 + 2} \times 2

Add the numbers

y^{3} \times 2

Use the commutative property to reorder the terms

2 y^{3}

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

Factor out 2 y^{3} from the expression

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

Solution

2 y^{3} (y - 2)

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Find the roots

y_{1} = 0, y_{2} = 2

Evaluate

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

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

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

Multiply the terms

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

Any expression multiplied by 1 remains the same

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

Calculate

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

Multiply

More Steps

Multiply the terms

y \times y^{2} \times 2

Multiply the terms with the same base by adding their exponents

y^{1 + 2} \times 2

Add the numbers

y^{3} \times 2

Use the commutative property to reorder the terms

2 y^{3}

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

Calculate

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Evaluate

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

Expand the expression

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Calculate

2 y^{3} (y - 1)

Apply the distributive property

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

Multiply the terms

2 y^{4} - 2 y^{3} \times 1

Any expression multiplied by 1 remains the same

2 y^{4} - 2 y^{3}

2 y^{4} - 2 y^{3} - 2 y^{3}

Subtract the terms

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Evaluate

- 2 y^{3} - 2 y^{3}

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

(- 2 - 2) y^{3}

Subtract the numbers

- 4 y^{3}

2 y^{4} - 4 y^{3}

2 y^{4} - 4 y^{3} = 0

Factor the expression

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

Divide both sides

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

Separate the equation into 2 possible cases

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

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

y = 0 y - 2 = 0

Solve the equation

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Evaluate

y - 2 = 0

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

y = 0 + 2

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

y = 2

y = 0 y = 2

Solution

y_{1} = 0, y_{2} = 2

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