Solve (a^2)(a-3)(a-2)(a^3)

Question

Simplify the expression

a^{7} - 5 a^{6} + 6 a^{5}

Evaluate

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

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

Multiply the terms

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Evaluate

a^{5} (a - 3)

Apply the distributive property

a^{5} \times a - a^{5} \times 3

Multiply the terms

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Evaluate

a^{5} \times a

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

a^{5 + 1}

Add the numbers

a^{6}

a^{6} - a^{5} \times 3

Use the commutative property to reorder the terms

a^{6} - 3 a^{5}

(a^{6} - 3 a^{5}) (a - 2)

Apply the distributive property

a^{6} \times a - a^{6} \times 2 - 3 a^{5} \times a - (- 3 a^{5} \times 2)

Multiply the terms

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Evaluate

a^{6} \times a

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

a^{6 + 1}

Add the numbers

a^{7}

a^{7} - a^{6} \times 2 - 3 a^{5} \times a - (- 3 a^{5} \times 2)

Use the commutative property to reorder the terms

a^{7} - 2 a^{6} - 3 a^{5} \times a - (- 3 a^{5} \times 2)

Multiply the terms

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Evaluate

a^{5} \times a

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

a^{5 + 1}

Add the numbers

a^{6}

a^{7} - 2 a^{6} - 3 a^{6} - (- 3 a^{5} \times 2)

Multiply the numbers

a^{7} - 2 a^{6} - 3 a^{6} - (- 6 a^{5})

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^{7} - 2 a^{6} - 3 a^{6} + 6 a^{5}

Solution

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Evaluate

- 2 a^{6} - 3 a^{6}

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

(- 2 - 3) a^{6}

Subtract the numbers

- 5 a^{6}

a^{7} - 5 a^{6} + 6 a^{5}

Show Solution

Find the roots

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

Evaluate

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

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

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

Calculate

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

Calculate

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

Multiply

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Multiply the terms

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

Multiply the terms with the same base by adding their exponents

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

Add the numbers

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

a^{5} (a - 3) (a - 2) = 0

Separate the equation into 3 possible cases

a^{5} = 0 a - 3 = 0 a - 2 = 0

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

a = 0 a - 3 = 0 a - 2 = 0

Solve the equation

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Evaluate

a - 3 = 0

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

a = 0 + 3

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

a = 3

a = 0 a = 3 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 = 3 a = 2

Solution

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

Show Solution