Solve 5Z^2 3Z 1 Z(Z - 1)(Z - 2)

Question

Simplify the expression

Solution

15 Z^{6} - 45 Z^{5} + 30 Z^{4}

Evaluate

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

Rewrite the expression

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

Multiply the terms

15 Z^{2} \times Z \times Z (Z - 1) (Z - 2)

Multiply the terms with the same base by adding their exponents

15 Z^{2 + 1 + 1} (Z - 1) (Z - 2)

Add the numbers

15 Z^{4} (Z - 1) (Z - 2)

Multiply the terms

More Steps

Evaluate

15 Z^{4} (Z - 1)

Apply the distributive property

15 Z^{4} \times Z - 15 Z^{4} \times 1

Multiply the terms

More Steps

Evaluate

Z^{4} \times Z

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

Z^{4 + 1}

Add the numbers

Z^{5}

15 Z^{5} - 15 Z^{4} \times 1

Any expression multiplied by 1 remains the same

15 Z^{5} - 15 Z^{4}

(15 Z^{5} - 15 Z^{4}) (Z - 2)

Apply the distributive property

15 Z^{5} \times Z - 15 Z^{5} \times 2 - 15 Z^{4} \times Z - (- 15 Z^{4} \times 2)

Multiply the terms

More Steps

Evaluate

Z^{5} \times Z

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

Z^{5 + 1}

Add the numbers

Z^{6}

15 Z^{6} - 15 Z^{5} \times 2 - 15 Z^{4} \times Z - (- 15 Z^{4} \times 2)

Multiply the numbers

15 Z^{6} - 30 Z^{5} - 15 Z^{4} \times Z - (- 15 Z^{4} \times 2)

Multiply the terms

More Steps

Evaluate

Z^{4} \times Z

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

Z^{4 + 1}

Add the numbers

Z^{5}

15 Z^{6} - 30 Z^{5} - 15 Z^{5} - (- 15 Z^{4} \times 2)

Multiply the numbers

15 Z^{6} - 30 Z^{5} - 15 Z^{5} - (- 30 Z^{4})

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

15 Z^{6} - 30 Z^{5} - 15 Z^{5} + 30 Z^{4}

Solution

More Steps

Evaluate

- 30 Z^{5} - 15 Z^{5}

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

(- 30 - 15) Z^{5}

Subtract the numbers

- 45 Z^{5}

15 Z^{6} - 45 Z^{5} + 30 Z^{4}

Show Solution

Find the roots

Find the roots of the algebra expression

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

Evaluate

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

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

5 Z^{2} \times 3 Z \times 1 \times Z (Z - 1) (Z - 2) = 0

Multiply the terms

More Steps

Multiply the terms

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

Rewrite the expression

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

Multiply the terms

15 Z^{2} \times Z \times Z (Z - 1) (Z - 2)

Multiply the terms with the same base by adding their exponents

15 Z^{2 + 1 + 1} (Z - 1) (Z - 2)

Add the numbers

15 Z^{4} (Z - 1) (Z - 2)

15 Z^{4} (Z - 1) (Z - 2) = 0

Elimination the left coefficient

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

Separate the equation into 3 possible cases

Z^{4} = 0 Z - 1 = 0 Z - 2 = 0

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

Z = 0 Z - 1 = 0 Z - 2 = 0

Solve the equation

More Steps

Evaluate

Z - 1 = 0

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

Z = 0 + 1

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

Z = 1

Z = 0 Z = 1 Z - 2 = 0

Solve the equation

More Steps

Evaluate

Z - 2 = 0

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

Z = 0 + 2

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

Z = 2

Z = 0 Z = 1 Z = 2

Solution

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

Show Solution

5z^2 3z 1 z(z 1)(z 2)

Simplify the expression

Solution

Find the roots

Find the roots of the algebra expression