Algebra Examples

$5 w (- 7 w + 3) + 2 w (- 2 w^{2} + 19 w + 2)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$5 w (- 7 w) + 5 w \cdot 3 + 2 w (- 2 w^{2} + 19 w + 2)$

Step 1.2

Rewrite using the commutative property of multiplication.

$5 \cdot - 7 w \cdot w + 5 w \cdot 3 + 2 w (- 2 w^{2} + 19 w + 2)$

Step 1.3

Multiply $3$ by $5$ .

$5 \cdot - 7 w \cdot w + 15 w + 2 w (- 2 w^{2} + 19 w + 2)$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $w$ by $w$ by adding the exponents.

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Step 1.4.1.1

Move $w$ .

$5 \cdot - 7 (w \cdot w) + 15 w + 2 w (- 2 w^{2} + 19 w + 2)$

Step 1.4.1.2

Multiply $w$ by $w$ .

$5 \cdot - 7 w^{2} + 15 w + 2 w (- 2 w^{2} + 19 w + 2)$

Step 1.4.2

Multiply $5$ by $- 7$ .

$- 35 w^{2} + 15 w + 2 w (- 2 w^{2} + 19 w + 2)$

Step 1.5

Apply the distributive property.

$- 35 w^{2} + 15 w + 2 w (- 2 w^{2}) + 2 w (19 w) + 2 w \cdot 2$

Step 1.6

Simplify.

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Step 1.6.1

Rewrite using the commutative property of multiplication.

$- 35 w^{2} + 15 w + 2 \cdot - 2 w \cdot w^{2} + 2 w (19 w) + 2 w \cdot 2$

Step 1.6.2

Rewrite using the commutative property of multiplication.

$- 35 w^{2} + 15 w + 2 \cdot - 2 w \cdot w^{2} + 2 \cdot 19 w \cdot w + 2 w \cdot 2$

Step 1.6.3

Multiply $2$ by $2$ .

$- 35 w^{2} + 15 w + 2 \cdot - 2 w \cdot w^{2} + 2 \cdot 19 w \cdot w + 4 w$

Step 1.7

Simplify each term.

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Step 1.7.1

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

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Step 1.7.1.1

Move $w^{2}$ .

$- 35 w^{2} + 15 w + 2 \cdot - 2 (w^{2} w) + 2 \cdot 19 w \cdot w + 4 w$

Step 1.7.1.2

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

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Step 1.7.1.2.1

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

$- 35 w^{2} + 15 w + 2 \cdot - 2 (w^{2} w^{1}) + 2 \cdot 19 w \cdot w + 4 w$

Step 1.7.1.2.2

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

$- 35 w^{2} + 15 w + 2 \cdot - 2 w^{2 + 1} + 2 \cdot 19 w \cdot w + 4 w$

Step 1.7.1.3

Add $2$ and $1$ .

$- 35 w^{2} + 15 w + 2 \cdot - 2 w^{3} + 2 \cdot 19 w \cdot w + 4 w$

Step 1.7.2

Multiply $2$ by $- 2$ .

$- 35 w^{2} + 15 w - 4 w^{3} + 2 \cdot 19 w \cdot w + 4 w$

Step 1.7.3

Multiply $w$ by $w$ by adding the exponents.

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Step 1.7.3.1

Move $w$ .

$- 35 w^{2} + 15 w - 4 w^{3} + 2 \cdot 19 (w \cdot w) + 4 w$

Step 1.7.3.2

Multiply $w$ by $w$ .

$- 35 w^{2} + 15 w - 4 w^{3} + 2 \cdot 19 w^{2} + 4 w$

Step 1.7.4

Multiply $2$ by $19$ .

$- 35 w^{2} + 15 w - 4 w^{3} + 38 w^{2} + 4 w$

Step 2

Simplify by adding terms.

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Step 2.1

Add $- 35 w^{2}$ and $38 w^{2}$ .

$3 w^{2} + 15 w - 4 w^{3} + 4 w$

Step 2.2

Add $15 w$ and $4 w$ .

$3 w^{2} - 4 w^{3} + 19 w$

Step 2.3

Reorder $3 w^{2}$ and $- 4 w^{3}$ .

$- 4 w^{3} + 3 w^{2} + 19 w$