Solve (4k-4)(-4k^2-k-1)

Question

Simplify the expression

- 16 k^{3} + 12 k^{2} + 4

Evaluate

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

Apply the distributive property

4 k (- 4 k^{2}) - 4 k \times k - 4 k \times 1 - 4 (- 4 k^{2}) - (- 4 k) - (- 4 \times 1)

Multiply the terms

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Evaluate

4 k (- 4 k^{2})

Multiply the numbers

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Evaluate

4 (- 4)

Multiplying or dividing an odd number of negative terms equals a negative

- 4 \times 4

Multiply the numbers

- 16

- 16 k \times k^{2}

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}

- 16 k^{3}

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

Multiply the terms

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

Any expression multiplied by 1 remains the same

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

Multiply the numbers

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Evaluate

- 4 (- 4)

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

4 \times 4

Multiply the numbers

16

- 16 k^{3} - 4 k^{2} - 4 k + 16 k^{2} - (- 4 k) - (- 4 \times 1)

Any expression multiplied by 1 remains the same

- 16 k^{3} - 4 k^{2} - 4 k + 16 k^{2} - (- 4 k) - (- 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

- 16 k^{3} - 4 k^{2} - 4 k + 16 k^{2} + 4 k + 4

Add the terms

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Evaluate

- 4 k^{2} + 16 k^{2}

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

(- 4 + 16) k^{2}

Add the numbers

12 k^{2}

- 16 k^{3} + 12 k^{2} - 4 k + 4 k + 4

The sum of two opposites equals 0

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Evaluate

- 4 k + 4 k

Collect like terms

(- 4 + 4) k

Add the coefficients

0 \times k

Calculate

0

- 16 k^{3} + 12 k^{2} + 0 + 4

Solution

- 16 k^{3} + 12 k^{2} + 4

Show Solution

Factor the expression

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

Evaluate

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

Factor the expression

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

Factor the expression

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

Solution

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

Show Solution

Find the roots

k_{1} = - \frac{1}{8} - \frac{15}{8} i, k_{2} = - \frac{1}{8} + \frac{15}{8} i, k_{3} = 1

Alternative Form

k_{1} \approx - 0.125 - 0.484123 i, k_{2} \approx - 0.125 + 0.484123 i, k_{3} = 1

Evaluate

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

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

(4 k - 4) (- 4 k^{2} - k - 1) = 0

Separate the equation into 2 possible cases

4 k - 4 = 0 - 4 k^{2} - k - 1 = 0

Solve the equation

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Evaluate

4 k - 4 = 0

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

4 k = 0 + 4

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

4 k = 4

Divide both sides

\frac{4 k}{4} = \frac{4}{4}

Divide the numbers

k = \frac{4}{4}

Divide the numbers

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Evaluate

\frac{4}{4}

Reduce the numbers

\frac{1}{1}

Calculate

1

k = 1

k = 1 - 4 k^{2} - k - 1 = 0

Solve the equation

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Evaluate

- 4 k^{2} - k - 1 = 0

Multiply both sides

4 k^{2} + k + 1 = 0

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

k = \frac{- 1 \pm 1 ^{2} - 4 \times 4}{2 \times 4}

Simplify the expression

k = \frac{- 1 \pm 1 ^{2} - 4 \times 4}{8}

Simplify the expression

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Evaluate

1^{2} - 4 \times 4

1 raised to any power equals to 1

1 - 4 \times 4

Multiply the numbers

1 - 16

Subtract the numbers

- 15

k = \frac{- 1 \pm - 15}{8}

Simplify the radical expression

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Evaluate

- 15

Evaluate the power

15 \times - 1

Evaluate the power

15 \times i

k = \frac{- 1 \pm 15 \times i}{8}

Separate the equation into 2 possible cases

k = \frac{- 1 + 15 \times i}{8} k = \frac{- 1 - 15 \times i}{8}

Simplify the expression

k = - \frac{1}{8} + \frac{15}{8} i k = \frac{- 1 - 15 \times i}{8}

Simplify the expression

k = - \frac{1}{8} + \frac{15}{8} i k = - \frac{1}{8} - \frac{15}{8} i

k = 1 k = - \frac{1}{8} + \frac{15}{8} i k = - \frac{1}{8} - \frac{15}{8} i

Solution

k_{1} = - \frac{1}{8} - \frac{15}{8} i, k_{2} = - \frac{1}{8} + \frac{15}{8} i, k_{3} = 1

Alternative Form

k_{1} \approx - 0.125 - 0.484123 i, k_{2} \approx - 0.125 + 0.484123 i, k_{3} = 1

Show Solution