Algebra Examples

$(y - 1) (y - 2) (y - 3) = y^{3} - 3 y^{2} + 2 y$

Step 1

Simplify $(y - 1) (y - 2) (y - 3)$ .

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Step 1.1

Rewrite.

$0 + 0 + (y - 1) (y - 2) (y - 3) = y^{3} - 3 y^{2} + 2 y$

Step 1.2

Simplify by adding zeros.

$(y - 1) (y - 2) (y - 3) = y^{3} - 3 y^{2} + 2 y$

Step 1.3

Expand $(y - 1) (y - 2)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$(y (y - 2) - 1 (y - 2)) (y - 3) = y^{3} - 3 y^{2} + 2 y$

Step 1.3.2

Apply the distributive property.

$(y \cdot y + y \cdot - 2 - 1 (y - 2)) (y - 3) = y^{3} - 3 y^{2} + 2 y$

Step 1.3.3

Apply the distributive property.

$(y \cdot y + y \cdot - 2 - 1 y - 1 \cdot - 2) (y - 3) = y^{3} - 3 y^{2} + 2 y$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $y$ by $y$ .

$(y^{2} + y \cdot - 2 - 1 y - 1 \cdot - 2) (y - 3) = y^{3} - 3 y^{2} + 2 y$

Step 1.4.1.2

Move $- 2$ to the left of $y$ .

$(y^{2} - 2 \cdot y - 1 y - 1 \cdot - 2) (y - 3) = y^{3} - 3 y^{2} + 2 y$

Step 1.4.1.3

Rewrite $- 1 y$ as $- y$ .

$(y^{2} - 2 y - y - 1 \cdot - 2) (y - 3) = y^{3} - 3 y^{2} + 2 y$

Step 1.4.1.4

Multiply $- 1$ by $- 2$ .

$(y^{2} - 2 y - y + 2) (y - 3) = y^{3} - 3 y^{2} + 2 y$

Step 1.4.2

Subtract $y$ from $- 2 y$ .

$(y^{2} - 3 y + 2) (y - 3) = y^{3} - 3 y^{2} + 2 y$

Step 1.5

Expand $(y^{2} - 3 y + 2) (y - 3)$ by multiplying each term in the first expression by each term in the second expression.

$y^{2} y + y^{2} \cdot - 3 - 3 y \cdot y - 3 y \cdot - 3 + 2 y + 2 \cdot - 3 = y^{3} - 3 y^{2} + 2 y$

Step 1.6

Simplify terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 1.6.1.1.1

Multiply $y^{2}$ by $y$ .

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Step 1.6.1.1.1.1

Raise $y$ to the power of $1$ .

$y^{2} y^{1} + y^{2} \cdot - 3 - 3 y \cdot y - 3 y \cdot - 3 + 2 y + 2 \cdot - 3 = y^{3} - 3 y^{2} + 2 y$

Step 1.6.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2 + 1} + y^{2} \cdot - 3 - 3 y \cdot y - 3 y \cdot - 3 + 2 y + 2 \cdot - 3 = y^{3} - 3 y^{2} + 2 y$

Step 1.6.1.1.2

Add $2$ and $1$ .

$y^{3} + y^{2} \cdot - 3 - 3 y \cdot y - 3 y \cdot - 3 + 2 y + 2 \cdot - 3 = y^{3} - 3 y^{2} + 2 y$

Step 1.6.1.2

Move $- 3$ to the left of $y^{2}$ .

$y^{3} - 3 \cdot y^{2} - 3 y \cdot y - 3 y \cdot - 3 + 2 y + 2 \cdot - 3 = y^{3} - 3 y^{2} + 2 y$

Step 1.6.1.3

Multiply $y$ by $y$ by adding the exponents.

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Step 1.6.1.3.1

Move $y$ .

$y^{3} - 3 y^{2} - 3 (y \cdot y) - 3 y \cdot - 3 + 2 y + 2 \cdot - 3 = y^{3} - 3 y^{2} + 2 y$

Step 1.6.1.3.2

Multiply $y$ by $y$ .

$y^{3} - 3 y^{2} - 3 y^{2} - 3 y \cdot - 3 + 2 y + 2 \cdot - 3 = y^{3} - 3 y^{2} + 2 y$

Step 1.6.1.4

Multiply $- 3$ by $- 3$ .

$y^{3} - 3 y^{2} - 3 y^{2} + 9 y + 2 y + 2 \cdot - 3 = y^{3} - 3 y^{2} + 2 y$

Step 1.6.1.5

Multiply $2$ by $- 3$ .

$y^{3} - 3 y^{2} - 3 y^{2} + 9 y + 2 y - 6 = y^{3} - 3 y^{2} + 2 y$

Step 1.6.2

Simplify by adding terms.

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Step 1.6.2.1

Subtract $3 y^{2}$ from $- 3 y^{2}$ .

$y^{3} - 6 y^{2} + 9 y + 2 y - 6 = y^{3} - 3 y^{2} + 2 y$

Step 1.6.2.2

Add $9 y$ and $2 y$ .

$y^{3} - 6 y^{2} + 11 y - 6 = y^{3} - 3 y^{2} + 2 y$

Step 2

Move all terms containing $y$ to the left side of the equation.

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Step 2.1

Subtract $y^{3}$ from both sides of the equation.

$y^{3} - 6 y^{2} + 11 y - 6 - y^{3} = - 3 y^{2} + 2 y$

Step 2.2

Add $3 y^{2}$ to both sides of the equation.

$y^{3} - 6 y^{2} + 11 y - 6 - y^{3} + 3 y^{2} = 2 y$

Step 2.3

Subtract $2 y$ from both sides of the equation.

$y^{3} - 6 y^{2} + 11 y - 6 - y^{3} + 3 y^{2} - 2 y = 0$

Step 2.4

Combine the opposite terms in $y^{3} - 6 y^{2} + 11 y - 6 - y^{3} + 3 y^{2} - 2 y$ .

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Step 2.4.1

Subtract $y^{3}$ from $y^{3}$ .

$- 6 y^{2} + 11 y - 6 + 0 + 3 y^{2} - 2 y = 0$

Step 2.4.2

Add $- 6 y^{2} + 11 y - 6$ and $0$ .

$- 6 y^{2} + 11 y - 6 + 3 y^{2} - 2 y = 0$

Step 2.5

Add $- 6 y^{2}$ and $3 y^{2}$ .

$- 3 y^{2} + 11 y - 6 - 2 y = 0$

Step 2.6

Subtract $2 y$ from $11 y$ .

$- 3 y^{2} + 9 y - 6 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- 3$ out of $- 3 y^{2} + 9 y - 6$ .

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Step 3.1.1

Factor $- 3$ out of $- 3 y^{2}$ .

$- 3 y^{2} + 9 y - 6 = 0$

Step 3.1.2

Factor $- 3$ out of $9 y$ .

$- 3 y^{2} - 3 (- 3 y) - 6 = 0$

Step 3.1.3

Factor $- 3$ out of $- 6$ .

$- 3 y^{2} - 3 (- 3 y) - 3 \cdot 2 = 0$

Step 3.1.4

Factor $- 3$ out of $- 3 (y^{2}) - 3 (- 3 y)$ .

$- 3 (y^{2} - 3 y) - 3 \cdot 2 = 0$

Step 3.1.5

Factor $- 3$ out of $- 3 (y^{2} - 3 y) - 3 (2)$ .

$- 3 (y^{2} - 3 y + 2) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor $y^{2} - 3 y + 2$ using the AC method.

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Step 3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 3.2.1.2

Write the factored form using these integers.

$- 3 ((y - 2) (y - 1)) = 0$

Step 3.2.2

Remove unnecessary parentheses.

$- 3 (y - 2) (y - 1) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 2 = 0$

$y - 1 = 0$

Step 5

Set $y - 2$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$y = 2$

Step 6

Set $y - 1$ equal to $0$ and solve for $y$ .

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Step 6.1

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

Step 6.2

Add $1$ to both sides of the equation.

$y = 1$

Step 7

The final solution is all the values that make $- 3 (y - 2) (y - 1) = 0$ true.

$y = 2, 1$