Solve (-4x-5)(-x^2 -x^4)

Question

Simplify the expression

4 x^{3} + 4 x^{5} + 5 x^{2} + 5 x^{4}

Evaluate

(- 4 x - 5) (- x^{2} - x^{4})

Apply the distributive property

- 4 x (- x^{2}) - (- 4 x \times x^{4}) - 5 (- x^{2}) - (- 5 x^{4})

Multiply the terms

More Steps

Evaluate

- 4 x (- x^{2})

Multiply the numbers

4 x \times x^{2}

Multiply the terms

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Evaluate

x \times x^{2}

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

x^{1 + 2}

Add the numbers

x^{3}

4 x^{3}

4 x^{3} - (- 4 x \times x^{4}) - 5 (- x^{2}) - (- 5 x^{4})

Multiply the terms

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Evaluate

x \times x^{4}

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

x^{1 + 4}

Add the numbers

x^{5}

4 x^{3} - (- 4 x^{5}) - 5 (- x^{2}) - (- 5 x^{4})

Multiply the numbers

4 x^{3} - (- 4 x^{5}) + 5 x^{2} - (- 5 x^{4})

Solution

4 x^{3} + 4 x^{5} + 5 x^{2} + 5 x^{4}

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Factor the expression

x^{2} (4 x + 5) (1 + x^{2})

Evaluate

(- 4 x - 5) (- x^{2} - x^{4})

Factor the expression

- (4 x + 5) (- x^{2} - x^{4})

Factor the expression

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Evaluate

- x^{2} - x^{4}

Rewrite the expression

- x^{2} - x^{2} \times x^{2}

Factor out - x^{2} from the expression

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

- (4 x + 5) (- x^{2}) (1 + x^{2})

Solution

x^{2} (4 x + 5) (1 + x^{2})

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Find the roots

x_{1} = - i, x_{2} = i, x_{3} = - \frac{5}{4}, x_{4} = 0

Alternative Form

x_{1} = - i, x_{2} = i, x_{3} = - 1.25, x_{4} = 0

Evaluate

(- 4 x - 5) (- x^{2} - x^{4})

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

(- 4 x - 5) (- x^{2} - x^{4}) = 0

Change the sign

(4 x + 5) (- x^{2} - x^{4}) = 0

Separate the equation into 2 possible cases

4 x + 5 = 0 - x^{2} - x^{4} = 0

Solve the equation

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Evaluate

4 x + 5 = 0

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

4 x = 0 - 5

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

4 x = - 5

Divide both sides

\frac{4 x}{4} = \frac{- 5}{4}

Divide the numbers

x = \frac{- 5}{4}

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

x = - \frac{5}{4}

x = - \frac{5}{4} - x^{2} - x^{4} = 0

Solve the equation

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Evaluate

- x^{2} - x^{4} = 0

Factor the expression

- x^{2} (1 + x^{2}) = 0

Divide both sides

x^{2} (1 + x^{2}) = 0

Separate the equation into 2 possible cases

x^{2} = 0 1 + x^{2} = 0

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

x = 0 1 + x^{2} = 0

Solve the equation

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Evaluate

1 + x^{2} = 0

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

x^{2} = 0 - 1

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

x^{2} = - 1

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm - 1

Simplify the expression

x = \pm i

Separate the equation into 2 possible cases

x = i x = - i

x = 0 x = i x = - i

x = - \frac{5}{4} x = 0 x = i x = - i

Solution

x_{1} = - i, x_{2} = i, x_{3} = - \frac{5}{4}, x_{4} = 0

Alternative Form

x_{1} = - i, x_{2} = i, x_{3} = - 1.25, x_{4} = 0

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