Solve (-7p^26p - 2) (5p^2 - 5p 8)

Question

Simplify the expression

- 210 p^{5} + 1680 p^{4} - 10 p^{2} + 80 p

Evaluate

(- 7 p^{2} \times 6 p - 2) (5 p^{2} - 5 p \times 8)

Multiply

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Multiply the terms

- 7 p^{2} \times 6 p

Multiply the terms

- 42 p^{2} \times p

Multiply the terms with the same base by adding their exponents

- 42 p^{2 + 1}

Add the numbers

- 42 p^{3}

(- 42 p^{3} - 2) (5 p^{2} - 5 p \times 8)

Multiply the terms

(- 42 p^{3} - 2) (5 p^{2} - 40 p)

Apply the distributive property

- 42 p^{3} \times 5 p^{2} - (- 42 p^{3} \times 40 p) - 2 \times 5 p^{2} - (- 2 \times 40 p)

Multiply the terms

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Evaluate

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

Multiply the numbers

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

Multiply the terms

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Evaluate

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

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

p^{3 + 2}

Add the numbers

p^{5}

- 210 p^{5}

- 210 p^{5} - (- 42 p^{3} \times 40 p) - 2 \times 5 p^{2} - (- 2 \times 40 p)

Multiply the terms

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Evaluate

- 42 p^{3} \times 40 p

Multiply the numbers

- 1680 p^{3} \times p

Multiply the terms

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Evaluate

p^{3} \times p

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

p^{3 + 1}

Add the numbers

p^{4}

- 1680 p^{4}

- 210 p^{5} - (- 1680 p^{4}) - 2 \times 5 p^{2} - (- 2 \times 40 p)

Multiply the numbers

- 210 p^{5} - (- 1680 p^{4}) - 10 p^{2} - (- 2 \times 40 p)

Multiply the numbers

- 210 p^{5} - (- 1680 p^{4}) - 10 p^{2} - (- 80 p)

Solution

- 210 p^{5} + 1680 p^{4} - 10 p^{2} + 80 p

Show Solution

Factor the expression

- 10 p (21 p^{3} + 1) (p - 8)

Evaluate

(- 7 p^{2} \times 6 p - 2) (5 p^{2} - 5 p \times 8)

Multiply

More Steps

Multiply the terms

- 7 p^{2} \times 6 p

Multiply the terms

- 42 p^{2} \times p

Multiply the terms with the same base by adding their exponents

- 42 p^{2 + 1}

Add the numbers

- 42 p^{3}

(- 42 p^{3} - 2) (5 p^{2} - 5 p \times 8)

Multiply the terms

(- 42 p^{3} - 2) (5 p^{2} - 40 p)

Factor the expression

- 2 (21 p^{3} + 1) (5 p^{2} - 40 p)

Factor the expression

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Evaluate

5 p^{2} - 40 p

Rewrite the expression

5 p \times p - 5 p \times 8

Factor out 5 p from the expression

5 p (p - 8)

- 2 (21 p^{3} + 1) \times 5 p (p - 8)

Solution

- 10 p (21 p^{3} + 1) (p - 8)

Show Solution

Find the roots

p_{1} = - \frac{3 441}{21}, p_{2} = 0, p_{3} = 8

Alternative Form

p_{1} \approx - 0.36246, p_{2} = 0, p_{3} = 8

Evaluate

(- 7 p^{2} \times 6 p - 2) (5 p^{2} - 5 p \times 8)

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

(- 7 p^{2} \times 6 p - 2) (5 p^{2} - 5 p \times 8) = 0

Multiply

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Multiply the terms

- 7 p^{2} \times 6 p

Multiply the terms

- 42 p^{2} \times p

Multiply the terms with the same base by adding their exponents

- 42 p^{2 + 1}

Add the numbers

- 42 p^{3}

(- 42 p^{3} - 2) (5 p^{2} - 5 p \times 8) = 0

Multiply the terms

(- 42 p^{3} - 2) (5 p^{2} - 40 p) = 0

Change the sign

(42 p^{3} + 2) (5 p^{2} - 40 p) = 0

Separate the equation into 2 possible cases

42 p^{3} + 2 = 0 5 p^{2} - 40 p = 0

Solve the equation

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Evaluate

42 p^{3} + 2 = 0

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

42 p^{3} = 0 - 2

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

42 p^{3} = - 2

Divide both sides

\frac{42 p ^{3}}{42} = \frac{- 2}{42}

Divide the numbers

p^{3} = \frac{- 2}{42}

Divide the numbers

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Evaluate

\frac{- 2}{42}

Cancel out the common factor 2

\frac{- 1}{21}

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

- \frac{1}{21}

p^{3} = - \frac{1}{21}

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

3 p^{3} = 3 - \frac{1}{21}

Calculate

p = 3 - \frac{1}{21}

Simplify the root

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Evaluate

3 - \frac{1}{21}

An odd root of a negative radicand is always a negative

- 3 \frac{1}{21}

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

- \frac{3 1}{3 21}

Simplify the radical expression

- \frac{1}{3 21}

Multiply by the Conjugate

\frac{- 3 2 1 ^{2}}{3 21 \times 3 2 1 ^{2}}

Simplify

\frac{- 3 441}{3 21 \times 3 2 1 ^{2}}

Multiply the numbers

\frac{- 3 441}{21}

Calculate

- \frac{3 441}{21}

p = - \frac{3 441}{21}

p = - \frac{3 441}{21} 5 p^{2} - 40 p = 0

Solve the equation

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Evaluate

5 p^{2} - 40 p = 0

Factor the expression

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Evaluate

5 p^{2} - 40 p

Rewrite the expression

5 p \times p - 5 p \times 8

Factor out 5 p from the expression

5 p (p - 8)

5 p (p - 8) = 0

When the product of factors equals 0,at least one factor is 0

5 p = 0 p - 8 = 0

Solve the equation for p

p = 0 p - 8 = 0

Solve the equation for p

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Evaluate

p - 8 = 0

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

p = 0 + 8

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

p = 8

p = 0 p = 8

p = - \frac{3 441}{21} p = 0 p = 8

Solution

p_{1} = - \frac{3 441}{21}, p_{2} = 0, p_{3} = 8

Alternative Form

p_{1} \approx - 0.36246, p_{2} = 0, p_{3} = 8

Show Solution