Solve k(k-1)^2 - k(k-1)

Question

Simplify the expression

k^{3} - 3 k^{2} + 2 k

Evaluate

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

Expand the expression

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Calculate

k (k - 1)^{2}

Simplify

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

Apply the distributive property

k \times k^{2} - k \times 2 k + k \times 1

Multiply the terms

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Evaluate

k \times k^{2}

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

k^{1 + 2}

Add the numbers

k^{3}

k^{3} - k \times 2 k + k \times 1

Multiply the terms

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Evaluate

k \times 2 k

Use the commutative property to reorder the terms

2 k \times k

Multiply the terms

2 k^{2}

k^{3} - 2 k^{2} + k \times 1

Any expression multiplied by 1 remains the same

k^{3} - 2 k^{2} + k

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

Expand the expression

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Calculate

- k (k - 1)

Apply the distributive property

- k \times k - (- k \times 1)

Multiply the terms

- k^{2} - (- k \times 1)

Any expression multiplied by 1 remains the same

- k^{2} - (- k)

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

- k^{2} + k

k^{3} - 2 k^{2} + k - k^{2} + k

Subtract the terms

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Evaluate

- 2 k^{2} - k^{2}

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

(- 2 - 1) k^{2}

Subtract the numbers

- 3 k^{2}

k^{3} - 3 k^{2} + k + k

Solution

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Evaluate

k + k

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

(1 + 1) k

Add the numbers

2 k

k^{3} - 3 k^{2} + 2 k

Show Solution

Factor the expression

k (k - 1) (k - 2)

Evaluate

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

Rewrite the expression

k (k - 1) (k - 1) - k (k - 1)

Factor out k (k - 1) from the expression

k (k - 1) (k - 1 - 1)

Solution

k (k - 1) (k - 2)

Show Solution

Find the roots

k_{1} = 0, k_{2} = 1, k_{3} = 2

Evaluate

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

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

k (k - 1)^{2} - k (k - 1) = 0

Calculate

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Evaluate

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

Expand the expression

More Steps

Calculate

k (k - 1)^{2}

Simplify

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

Apply the distributive property

k \times k^{2} - k \times 2 k + k \times 1

Multiply the terms

k^{3} - k \times 2 k + k \times 1

Multiply the terms

k^{3} - 2 k^{2} + k \times 1

Any expression multiplied by 1 remains the same

k^{3} - 2 k^{2} + k

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

Expand the expression

More Steps

Calculate

- k (k - 1)

Apply the distributive property

- k \times k - (- k \times 1)

Multiply the terms

- k^{2} - (- k \times 1)

Any expression multiplied by 1 remains the same

- k^{2} - (- k)

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

- k^{2} + k

k^{3} - 2 k^{2} + k - k^{2} + k

Subtract the terms

More Steps

Evaluate

- 2 k^{2} - k^{2}

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

(- 2 - 1) k^{2}

Subtract the numbers

- 3 k^{2}

k^{3} - 3 k^{2} + k + k

Add the terms

More Steps

Evaluate

k + k

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

(1 + 1) k

Add the numbers

2 k

k^{3} - 3 k^{2} + 2 k

k^{3} - 3 k^{2} + 2 k = 0

Factor the expression

k (k - 2) (k - 1) = 0

Separate the equation into 3 possible cases

k = 0 k - 2 = 0 k - 1 = 0

Solve the equation

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Evaluate

k - 2 = 0

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

k = 0 + 2

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

k = 2

k = 0 k = 2 k - 1 = 0

Solve the equation

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Evaluate

k - 1 = 0

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

k = 0 + 1

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

k = 1

k = 0 k = 2 k = 1

Solution

k_{1} = 0, k_{2} = 1, k_{3} = 2

Show Solution