Solve ((a^2) a-4)*((2a^2)-a-1)

Question

Simplify the expression

2 a^{5} - a^{4} - a^{3} - 8 a^{2} + 4 a + 4

Evaluate

(a^{2} \times a - 4) (2 a^{2} - a - 1)

Multiply the terms

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Evaluate

a^{2} \times a

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

a^{2 + 1}

Add the numbers

a^{3}

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

Apply the distributive property

a^{3} \times 2 a^{2} - a^{3} \times a - a^{3} \times 1 - 4 \times 2 a^{2} - (- 4 a) - (- 4 \times 1)

Multiply the terms

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Evaluate

a^{3} \times 2 a^{2}

Use the commutative property to reorder the terms

2 a^{3} \times a^{2}

Multiply the terms

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Evaluate

a^{3} \times a^{2}

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

a^{3 + 2}

Add the numbers

a^{5}

2 a^{5}

2 a^{5} - a^{3} \times a - a^{3} \times 1 - 4 \times 2 a^{2} - (- 4 a) - (- 4 \times 1)

Multiply the terms

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Evaluate

a^{3} \times a

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

a^{3 + 1}

Add the numbers

a^{4}

2 a^{5} - a^{4} - a^{3} \times 1 - 4 \times 2 a^{2} - (- 4 a) - (- 4 \times 1)

Any expression multiplied by 1 remains the same

2 a^{5} - a^{4} - a^{3} - 4 \times 2 a^{2} - (- 4 a) - (- 4 \times 1)

Multiply the numbers

2 a^{5} - a^{4} - a^{3} - 8 a^{2} - (- 4 a) - (- 4 \times 1)

Any expression multiplied by 1 remains the same

2 a^{5} - a^{4} - a^{3} - 8 a^{2} - (- 4 a) - (- 4)

Solution

2 a^{5} - a^{4} - a^{3} - 8 a^{2} + 4 a + 4

Show Solution

Factor the expression

(a^{3} - 4) (a - 1) (2 a + 1)

Evaluate

(a^{2} \times a - 4) (2 a^{2} - a - 1)

Multiply the terms

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Evaluate

a^{2} \times a

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

a^{2 + 1}

Add the numbers

a^{3}

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

Solution

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Evaluate

2 a^{2} - a - 1

Rewrite the expression

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

Calculate

2 a^{2} + a - 2 a - 1

Rewrite the expression

a \times 2 a + a - 2 a - 1

Factor out a from the expression

a (2 a + 1) - 2 a - 1

Factor out - 1 from the expression

a (2 a + 1) - (2 a + 1)

Factor out 2 a + 1 from the expression

(a - 1) (2 a + 1)

(a^{3} - 4) (a - 1) (2 a + 1)

Show Solution

Find the roots

a_{1} = - \frac{1}{2}, a_{2} = 1, a_{3} = 34

Alternative Form

a_{1} = - 0.5, a_{2} = 1, a_{3} \approx 1.587401

Evaluate

((a^{2}) a - 4) ((2 a^{2}) - a - 1)

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

((a^{2}) a - 4) ((2 a^{2}) - a - 1) = 0

Calculate

(a^{2} \times a - 4) ((2 a^{2}) - a - 1) = 0

Multiply the terms

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Evaluate

a^{2} \times a

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

a^{2 + 1}

Add the numbers

a^{3}

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

Multiply the terms

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

Separate the equation into 2 possible cases

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

Solve the equation

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Evaluate

a^{3} - 4 = 0

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

a^{3} = 0 + 4

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

a^{3} = 4

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

3 a^{3} = 34

Calculate

a = 34

a = 34 2 a^{2} - a - 1 = 0

Solve the equation

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Evaluate

2 a^{2} - a - 1 = 0

Factor the expression

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Evaluate

2 a^{2} - a - 1

Rewrite the expression

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

Calculate

2 a^{2} + a - 2 a - 1

Rewrite the expression

a \times 2 a + a - 2 a - 1

Factor out a from the expression

a (2 a + 1) - 2 a - 1

Factor out - 1 from the expression

a (2 a + 1) - (2 a + 1)

Factor out 2 a + 1 from the expression

(a - 1) (2 a + 1)

(a - 1) (2 a + 1) = 0

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

a - 1 = 0 2 a + 1 = 0

Solve the equation for a

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Evaluate

a - 1 = 0

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

a = 0 + 1

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

a = 1

a = 1 2 a + 1 = 0

Solve the equation for a

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Evaluate

2 a + 1 = 0

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

2 a = 0 - 1

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

2 a = - 1

Divide both sides

\frac{2 a}{2} = \frac{- 1}{2}

Divide the numbers

a = \frac{- 1}{2}

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

a = - \frac{1}{2}

a = 1 a = - \frac{1}{2}

a = 34 a = 1 a = - \frac{1}{2}

Solution

a_{1} = - \frac{1}{2}, a_{2} = 1, a_{3} = 34

Alternative Form

a_{1} = - 0.5, a_{2} = 1, a_{3} \approx 1.587401

Show Solution