Solve (2x^2-x-15)(2x^5)

Question

Simplify the expression

4 x^{7} - 2 x^{6} - 30 x^{5}

Evaluate

(2 x^{2} - x - 15) \times 2 x^{5}

Multiply the terms

2 x^{5} (2 x^{2} - x - 15)

Apply the distributive property

2 x^{5} \times 2 x^{2} - 2 x^{5} \times x - 2 x^{5} \times 15

Multiply the terms

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Evaluate

2 x^{5} \times 2 x^{2}

Multiply the numbers

4 x^{5} \times x^{2}

Multiply the terms

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Evaluate

x^{5} \times x^{2}

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

x^{5 + 2}

Add the numbers

x^{7}

4 x^{7}

4 x^{7} - 2 x^{5} \times x - 2 x^{5} \times 15

Multiply the terms

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Evaluate

x^{5} \times x

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

x^{5 + 1}

Add the numbers

x^{6}

4 x^{7} - 2 x^{6} - 2 x^{5} \times 15

Solution

4 x^{7} - 2 x^{6} - 30 x^{5}

Show Solution

Factor the expression

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

Evaluate

(2 x^{2} - x - 15) \times 2 x^{5}

Multiply the terms

2 x^{5} (2 x^{2} - x - 15)

Solution

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Evaluate

2 x^{2} - x - 15

Rewrite the expression

2 x^{2} + (5 - 6) x - 15

Calculate

2 x^{2} + 5 x - 6 x - 15

Rewrite the expression

x \times 2 x + x \times 5 - 3 \times 2 x - 3 \times 5

Factor out x from the expression

x (2 x + 5) - 3 \times 2 x - 3 \times 5

Factor out - 3 from the expression

x (2 x + 5) - 3 (2 x + 5)

Factor out 2 x + 5 from the expression

(x - 3) (2 x + 5)

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

Show Solution

Find the roots

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

Alternative Form

x_{1} = - 2.5, x_{2} = 0, x_{3} = 3

Evaluate

(2 x^{2} - x - 15) (2 x^{5})

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

(2 x^{2} - x - 15) (2 x^{5}) = 0

Multiply the terms

(2 x^{2} - x - 15) \times 2 x^{5} = 0

Multiply the terms

2 x^{5} (2 x^{2} - x - 15) = 0

Elimination the left coefficient

x^{5} (2 x^{2} - x - 15) = 0

Separate the equation into 2 possible cases

x^{5} = 0 2 x^{2} - x - 15 = 0

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

x = 0 2 x^{2} - x - 15 = 0

Solve the equation

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Evaluate

2 x^{2} - x - 15 = 0

Factor the expression

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Evaluate

2 x^{2} - x - 15

Rewrite the expression

2 x^{2} + (5 - 6) x - 15

Calculate

2 x^{2} + 5 x - 6 x - 15

Rewrite the expression

x \times 2 x + x \times 5 - 3 \times 2 x - 3 \times 5

Factor out x from the expression

x (2 x + 5) - 3 \times 2 x - 3 \times 5

Factor out - 3 from the expression

x (2 x + 5) - 3 (2 x + 5)

Factor out 2 x + 5 from the expression

(x - 3) (2 x + 5)

(x - 3) (2 x + 5) = 0

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

x - 3 = 0 2 x + 5 = 0

Solve the equation for x

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Evaluate

x - 3 = 0

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

x = 0 + 3

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

x = 3

x = 3 2 x + 5 = 0

Solve the equation for x

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Evaluate

2 x + 5 = 0

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

2 x = 0 - 5

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

2 x = - 5

Divide both sides

\frac{2 x}{2} = \frac{- 5}{2}

Divide the numbers

x = \frac{- 5}{2}

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

x = - \frac{5}{2}

x = 3 x = - \frac{5}{2}

x = 0 x = 3 x = - \frac{5}{2}

Solution

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

Alternative Form

x_{1} = - 2.5, x_{2} = 0, x_{3} = 3

Show Solution