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

Question

Simplify the expression

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

Evaluate

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

Remove the parentheses

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

Rewrite the expression

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

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

Multiply the terms

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Evaluate

a^{3} (a - 1)

Apply the distributive property

a^{3} \times a - a^{3} \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}

a^{4} - a^{3} \times 1

Any expression multiplied by 1 remains the same

a^{4} - a^{3}

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

Apply the distributive property

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

Multiply the terms

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Evaluate

a^{4} \times a

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

a^{4 + 1}

Add the numbers

a^{5}

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

Use the commutative property to reorder the terms

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

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}

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

Use the commutative property to reorder the terms

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

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

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

Solution

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Evaluate

- 2 a^{4} - a^{4}

Collect like terms by calculating the sum or difference of their coefficients

(- 2 - 1) a^{4}

Subtract the numbers

- 3 a^{4}

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

Show Solution

Find the roots

a_{1} = 0, a_{2} = 1, a_{3} = 2

Evaluate

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

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

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

Any expression multiplied by 1 remains the same

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

Calculate

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

Multiply

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Multiply the terms

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

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

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

Separate the equation into 3 possible cases

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

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

a = 0 a - 1 = 0 a - 2 = 0

Solve the equation

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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 = 0 a = 1 a - 2 = 0

Solve the equation

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Evaluate

a - 2 = 0

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

a = 0 + 2

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

a = 2

a = 0 a = 1 a = 2

Solution

a_{1} = 0, a_{2} = 1, a_{3} = 2

Show Solution