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

Question

Simplify the expression

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

Evaluate

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

Expand the expression

More Steps

Calculate

k (k - 1)^{3}

Simplify

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

Apply the distributive property

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

Multiply the terms

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Evaluate

k \times k^{3}

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

k^{1 + 3}

Add the numbers

k^{4}

k^{4} - k \times 3 k^{2} + k \times 3 k - k \times 1

Multiply the terms

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Evaluate

k \times 3 k^{2}

Use the commutative property to reorder the terms

3 k \times k^{2}

Multiply the terms

3 k^{3}

k^{4} - 3 k^{3} + k \times 3 k - k \times 1

Multiply the terms

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Evaluate

k \times 3 k

Use the commutative property to reorder the terms

3 k \times k

Multiply the terms

3 k^{2}

k^{4} - 3 k^{3} + 3 k^{2} - k \times 1

Any expression multiplied by 1 remains the same

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

k^{4} - 3 k^{3} + 3 k^{2} - k - k (k - 1) (k - 2)

Expand the expression

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Calculate

- k (k - 1) (k - 2)

Simplify

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Evaluate

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

Apply the distributive property

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

Multiply the terms

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Evaluate

k^{2} \times k

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

k^{2 + 1}

Add the numbers

k^{3}

- k^{3} - (- k^{2} \times 2) + k \times k - k \times 2

Use the commutative property to reorder the terms

- k^{3} - (- 2 k^{2}) + k \times k - k \times 2

Multiply the terms

- k^{3} - (- 2 k^{2}) + k^{2} - k \times 2

Use the commutative property to reorder the terms

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

Add 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}

Add the numbers

3 k^{2}

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

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

Subtract the terms

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Evaluate

- 3 k^{3} - k^{3}

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

(- 3 - 1) k^{3}

Subtract the numbers

- 4 k^{3}

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

Add the terms

More Steps

Evaluate

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

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

(3 + 3) k^{2}

Add the numbers

6 k^{2}

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

Solution

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Evaluate

- k - 2 k

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

(- 1 - 2) k

Subtract the numbers

- 3 k

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

Show Solution

Factor the expression

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

Evaluate

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

Rewrite the expression

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

Factor out k (k - 1) from the expression

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

Solution

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

Show Solution

Find the roots

k_{1} = \frac{3}{2} - \frac{3}{2} i, k_{2} = \frac{3}{2} + \frac{3}{2} i, k_{3} = 0, k_{4} = 1

Alternative Form

k_{1} \approx 1.5 - 0.866025 i, k_{2} \approx 1.5 + 0.866025 i, k_{3} = 0, k_{4} = 1

Evaluate

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

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

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

Calculate

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Evaluate

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

Expand the expression

More Steps

Calculate

k (k - 1)^{3}

Simplify

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

Apply the distributive property

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

Multiply the terms

k^{4} - k \times 3 k^{2} + k \times 3 k - k \times 1

Multiply the terms

k^{4} - 3 k^{3} + k \times 3 k - k \times 1

Multiply the terms

k^{4} - 3 k^{3} + 3 k^{2} - k \times 1

Any expression multiplied by 1 remains the same

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

k^{4} - 3 k^{3} + 3 k^{2} - k - k (k - 1) (k - 2)

Expand the expression

More Steps

Calculate

- k (k - 1) (k - 2)

Simplify

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

Apply the distributive property

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

Multiply the terms

- k^{3} - (- k^{2} \times 2) + k \times k - k \times 2

Use the commutative property to reorder the terms

- k^{3} - (- 2 k^{2}) + k \times k - k \times 2

Multiply the terms

- k^{3} - (- 2 k^{2}) + k^{2} - k \times 2

Use the commutative property to reorder the terms

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

Add the terms

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

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

Subtract the terms

More Steps

Evaluate

- 3 k^{3} - k^{3}

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

(- 3 - 1) k^{3}

Subtract the numbers

- 4 k^{3}

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

Add the terms

More Steps

Evaluate

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

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

(3 + 3) k^{2}

Add the numbers

6 k^{2}

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

Subtract the terms

More Steps

Evaluate

- k - 2 k

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

(- 1 - 2) k

Subtract the numbers

- 3 k

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

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

Factor the expression

k (k - 1) (k^{2} - 3 k + 3) = 0

Separate the equation into 3 possible cases

k = 0 k - 1 = 0 k^{2} - 3 k + 3 = 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 = 1 k^{2} - 3 k + 3 = 0

Solve the equation

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Evaluate

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

Substitute a= 1,b= - 3 and c= 3 into the quadratic formula k = \frac{- b \pm b ^{2} - 4 a c}{2 a}

k = \frac{3 \pm ( - 3 ) ^{2} - 4 \times 3}{2}

Simplify the expression

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Evaluate

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

Multiply the numbers

(- 3)^{2} - 12

Rewrite the expression

3^{2} - 12

Evaluate the power

9 - 12

Subtract the numbers

- 3

k = \frac{3 \pm - 3}{2}

Simplify the radical expression

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Evaluate

- 3

Evaluate the power

3 \times - 1

Evaluate the power

3 \times i

k = \frac{3 \pm 3 \times i}{2}

Separate the equation into 2 possible cases

k = \frac{3 + 3 \times i}{2} k = \frac{3 - 3 \times i}{2}

Simplify the expression

k = \frac{3}{2} + \frac{3}{2} i k = \frac{3 - 3 \times i}{2}

Simplify the expression

k = \frac{3}{2} + \frac{3}{2} i k = \frac{3}{2} - \frac{3}{2} i

k = 0 k = 1 k = \frac{3}{2} + \frac{3}{2} i k = \frac{3}{2} - \frac{3}{2} i

Solution

k_{1} = \frac{3}{2} - \frac{3}{2} i, k_{2} = \frac{3}{2} + \frac{3}{2} i, k_{3} = 0, k_{4} = 1

Alternative Form

k_{1} \approx 1.5 - 0.866025 i, k_{2} \approx 1.5 + 0.866025 i, k_{3} = 0, k_{4} = 1

Show Solution