Solve (4v-1)( - 4v^2-2v^3)

Question

Simplify the expression

- 14 v^{3} - 8 v^{4} + 4 v^{2}

Evaluate

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

Apply the distributive property

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

Multiply the terms

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Evaluate

4 v (- 4 v^{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 v \times v^{2}

Multiply the terms

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Evaluate

v \times v^{2}

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

v^{1 + 2}

Add the numbers

v^{3}

- 16 v^{3}

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

Multiply the terms

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Evaluate

4 v \times 2 v^{3}

Multiply the numbers

8 v \times v^{3}

Multiply the terms

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Evaluate

v \times v^{3}

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

v^{1 + 3}

Add the numbers

v^{4}

8 v^{4}

- 16 v^{3} - 8 v^{4} - (- 4 v^{2}) - (- 2 v^{3})

When there is - in front of an expression in parentheses change the sign of each term of the expression and remove the parentheses

- 16 v^{3} - 8 v^{4} + 4 v^{2} - (- 2 v^{3})

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 v^{3} - 8 v^{4} + 4 v^{2} + 2 v^{3}

Solution

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Evaluate

- 16 v^{3} + 2 v^{3}

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

(- 16 + 2) v^{3}

Add the numbers

- 14 v^{3}

- 14 v^{3} - 8 v^{4} + 4 v^{2}

Show Solution

Factor the expression

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

Evaluate

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

Factor the expression

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Evaluate

- 4 v^{2} - 2 v^{3}

Rewrite the expression

- 2 v^{2} \times 2 - 2 v^{2} \times v

Factor out - 2 v^{2} from the expression

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

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

Solution

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

Show Solution

Find the roots

v_{1} = - 2, v_{2} = 0, v_{3} = \frac{1}{4}

Alternative Form

v_{1} = - 2, v_{2} = 0, v_{3} = 0.25

Evaluate

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

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

(4 v - 1) (- 4 v^{2} - 2 v^{3}) = 0

Separate the equation into 2 possible cases

4 v - 1 = 0 - 4 v^{2} - 2 v^{3} = 0

Solve the equation

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Evaluate

4 v - 1 = 0

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

4 v = 0 + 1

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

4 v = 1

Divide both sides

\frac{4 v}{4} = \frac{1}{4}

Divide the numbers

v = \frac{1}{4}

v = \frac{1}{4} - 4 v^{2} - 2 v^{3} = 0

Solve the equation

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Evaluate

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

Factor the expression

- 2 v^{2} (2 + v) = 0

Divide both sides

v^{2} (2 + v) = 0

Separate the equation into 2 possible cases

v^{2} = 0 2 + v = 0

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

v = 0 2 + v = 0

Solve the equation

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Evaluate

2 + v = 0

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

v = 0 - 2

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

v = - 2

v = 0 v = - 2

v = \frac{1}{4} v = 0 v = - 2

Solution

v_{1} = - 2, v_{2} = 0, v_{3} = \frac{1}{4}

Alternative Form

v_{1} = - 2, v_{2} = 0, v_{3} = 0.25

Show Solution