Solve (-p^2 4p-3)(p^2 2)

Question

Simplify the expression

- 8 p^{5} - 6 p^{2}

Evaluate

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

Remove the parentheses

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

Multiply

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Multiply the terms

- p^{2} \times 4 p

Multiply the terms with the same base by adding their exponents

- p^{2 + 1} \times 4

Add the numbers

- p^{3} \times 4

Use the commutative property to reorder the terms

- 4 p^{3}

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

Use the commutative property to reorder the terms

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

Multiply the terms

2 p^{2} (- 4 p^{3} - 3)

Apply the distributive property

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

Multiply the terms

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Evaluate

2 p^{2} (- 4 p^{3})

Multiply the numbers

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Evaluate

2 (- 4)

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

- 2 \times 4

Multiply the numbers

- 8

- 8 p^{2} \times p^{3}

Multiply the terms

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Evaluate

p^{2} \times p^{3}

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

p^{2 + 3}

Add the numbers

p^{5}

- 8 p^{5}

- 8 p^{5} - 2 p^{2} \times 3

Solution

- 8 p^{5} - 6 p^{2}

Show Solution

Find the roots

p_{1} = - \frac{3 6}{2}, p_{2} = 0

Alternative Form

p_{1} \approx - 0.90856, p_{2} = 0

Evaluate

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

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

(- p^{2} \times 4 p - 3) (p^{2} \times 2) = 0

Multiply

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Multiply the terms

p^{2} \times 4 p

Multiply the terms with the same base by adding their exponents

p^{2 + 1} \times 4

Add the numbers

p^{3} \times 4

Use the commutative property to reorder the terms

4 p^{3}

(- 4 p^{3} - 3) (p^{2} \times 2) = 0

Use the commutative property to reorder the terms

(- 4 p^{3} - 3) \times 2 p^{2} = 0

Multiply the terms

2 p^{2} (- 4 p^{3} - 3) = 0

Elimination the left coefficient

p^{2} (- 4 p^{3} - 3) = 0

Separate the equation into 2 possible cases

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

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

p = 0 - 4 p^{3} - 3 = 0

Solve the equation

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Evaluate

- 4 p^{3} - 3 = 0

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

- 4 p^{3} = 0 + 3

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

- 4 p^{3} = 3

Change the signs on both sides of the equation

4 p^{3} = - 3

Divide both sides

\frac{4 p ^{3}}{4} = \frac{- 3}{4}

Divide the numbers

p^{3} = \frac{- 3}{4}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

p^{3} = - \frac{3}{4}

Take the 3 -th root on both sides of the equation

3 p^{3} = 3 - \frac{3}{4}

Calculate

p = 3 - \frac{3}{4}

Simplify the root

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Evaluate

3 - \frac{3}{4}

An odd root of a negative radicand is always a negative

- 3 \frac{3}{4}

To take a root of a fraction,take the root of the numerator and denominator separately

- \frac{3 3}{3 4}

Multiply by the Conjugate

\frac{- 3 3 \times 3 4 ^{2}}{3 4 \times 3 4 ^{2}}

Simplify

\frac{- 3 3 \times 2 3 2}{3 4 \times 3 4 ^{2}}

Multiply the numbers

\frac{- 2 3 6}{3 4 \times 3 4 ^{2}}

Multiply the numbers

\frac{- 2 3 6}{2 ^{2}}

Reduce the fraction

\frac{- 3 6}{2}

Calculate

- \frac{3 6}{2}

p = - \frac{3 6}{2}

p = 0 p = - \frac{3 6}{2}

Solution

p_{1} = - \frac{3 6}{2}, p_{2} = 0

Alternative Form

p_{1} \approx - 0.90856, p_{2} = 0

Show Solution