Algebra Examples

$(3 x - y) (6 x^{2} + 5 x y - 7 y^{2})$

Step 1

Expand $(3 x - y) (6 x^{2} + 5 x y - 7 y^{2})$ by multiplying each term in the first expression by each term in the second expression.

$3 x (6 x^{2}) + 3 x (5 x y) + 3 x (- 7 y^{2}) - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2

Simplify each term.

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Step 2.1

Rewrite using the commutative property of multiplication.

$3 \cdot 6 x \cdot x^{2} + 3 x (5 x y) + 3 x (- 7 y^{2}) - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.2

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

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Step 2.2.1

Move $x^{2}$ .

$3 \cdot 6 (x^{2} x) + 3 x (5 x y) + 3 x (- 7 y^{2}) - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.2.2

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

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Step 2.2.2.1

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

$3 \cdot 6 (x^{2} x^{1}) + 3 x (5 x y) + 3 x (- 7 y^{2}) - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.2.2.2

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

$3 \cdot 6 x^{2 + 1} + 3 x (5 x y) + 3 x (- 7 y^{2}) - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.2.3

Add $2$ and $1$ .

$3 \cdot 6 x^{3} + 3 x (5 x y) + 3 x (- 7 y^{2}) - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.3

Multiply $3$ by $6$ .

$18 x^{3} + 3 x (5 x y) + 3 x (- 7 y^{2}) - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.4

Multiply $x$ by $x$ by adding the exponents.

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Step 2.4.1

Move $x$ .

$18 x^{3} + 3 (x \cdot x) (5 y) + 3 x (- 7 y^{2}) - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.4.2

Multiply $x$ by $x$ .

$18 x^{3} + 3 x^{2} (5 y) + 3 x (- 7 y^{2}) - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.5

Rewrite using the commutative property of multiplication.

$18 x^{3} + 3 \cdot 5 x^{2} y + 3 x (- 7 y^{2}) - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.6

Multiply $3$ by $5$ .

$18 x^{3} + 15 x^{2} y + 3 x (- 7 y^{2}) - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.7

Rewrite using the commutative property of multiplication.

$18 x^{3} + 15 x^{2} y + 3 \cdot - 7 x y^{2} - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.8

Multiply $3$ by $- 7$ .

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - y (6 x^{2}) - y (5 x y) - y (- 7 y^{2})$

Step 2.9

Rewrite using the commutative property of multiplication.

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 1 \cdot 6 y x^{2} - y (5 x y) - y (- 7 y^{2})$

Step 2.10

Multiply $- 1$ by $6$ .

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 6 y x^{2} - y (5 x y) - y (- 7 y^{2})$

Step 2.11

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

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Step 2.11.1

Move $y$ .

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 6 y x^{2} - (y \cdot y) (5 x) - y (- 7 y^{2})$

Step 2.11.2

Multiply $y$ by $y$ .

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 6 y x^{2} - y^{2} (5 x) - y (- 7 y^{2})$

Step 2.12

Rewrite using the commutative property of multiplication.

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 6 y x^{2} - 1 \cdot 5 y^{2} x - y (- 7 y^{2})$

Step 2.13

Multiply $- 1$ by $5$ .

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 6 y x^{2} - 5 y^{2} x - y (- 7 y^{2})$

Step 2.14

Rewrite using the commutative property of multiplication.

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 6 y x^{2} - 5 y^{2} x - 1 \cdot - 7 y \cdot y^{2}$

Step 2.15

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

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Step 2.15.1

Move $y^{2}$ .

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 6 y x^{2} - 5 y^{2} x - 1 \cdot - 7 (y^{2} y)$

Step 2.15.2

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

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Step 2.15.2.1

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

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 6 y x^{2} - 5 y^{2} x - 1 \cdot - 7 (y^{2} y^{1})$

Step 2.15.2.2

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

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 6 y x^{2} - 5 y^{2} x - 1 \cdot - 7 y^{2 + 1}$

Step 2.15.3

Add $2$ and $1$ .

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 6 y x^{2} - 5 y^{2} x - 1 \cdot - 7 y^{3}$

Step 2.16

Multiply $- 1$ by $- 7$ .

$18 x^{3} + 15 x^{2} y - 21 x y^{2} - 6 y x^{2} - 5 y^{2} x + 7 y^{3}$

Step 3

Subtract $6 y x^{2}$ from $15 x^{2} y$ .

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Step 3.1

Move $y$ .

$18 x^{3} + 15 x^{2} y - 6 x^{2} y - 21 x y^{2} - 5 y^{2} x + 7 y^{3}$

Step 3.2

Subtract $6 x^{2} y$ from $15 x^{2} y$ .

$18 x^{3} + 9 x^{2} y - 21 x y^{2} - 5 y^{2} x + 7 y^{3}$

Step 4

Subtract $5 y^{2} x$ from $- 21 x y^{2}$ .

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Step 4.1

Move $y^{2}$ .

$18 x^{3} + 9 x^{2} y - 21 x y^{2} - 5 x y^{2} + 7 y^{3}$

Step 4.2

Subtract $5 x y^{2}$ from $- 21 x y^{2}$ .

$18 x^{3} + 9 x^{2} y - 26 x y^{2} + 7 y^{3}$