Expand the Trigonometric Expression (2+3w)(w^2+6w+9)

$(2 + 3 w) (w^{2} + 6 w + 9)$

Step 1

Expand $(2 + 3 w) (w^{2} + 6 w + 9)$ by multiplying each term in the first expression by each term in the second expression.

$2 w^{2} + 2 (6 w) + 2 \cdot 9 + 3 w \cdot w^{2} + 3 w (6 w) + 3 w \cdot 9$

Step 2

Simplify terms.

Simplify each term.

Multiply $6$ by $2$ .

$2 w^{2} + 12 w + 2 \cdot 9 + 3 w \cdot w^{2} + 3 w (6 w) + 3 w \cdot 9$

Multiply $2$ by $9$ .

$2 w^{2} + 12 w + 18 + 3 w \cdot w^{2} + 3 w (6 w) + 3 w \cdot 9$

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

Move $w^{2}$ .

$2 w^{2} + 12 w + 18 + 3 (w^{2} w) + 3 w (6 w) + 3 w \cdot 9$

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

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

$2 w^{2} + 12 w + 18 + 3 (w^{2} w^{1}) + 3 w (6 w) + 3 w \cdot 9$

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

$2 w^{2} + 12 w + 18 + 3 w^{2 + 1} + 3 w (6 w) + 3 w \cdot 9$

Add $2$ and $1$ .

$2 w^{2} + 12 w + 18 + 3 w^{3} + 3 w (6 w) + 3 w \cdot 9$

Rewrite using the commutative property of multiplication.

$2 w^{2} + 12 w + 18 + 3 w^{3} + 3 \cdot 6 w \cdot w + 3 w \cdot 9$

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

Move $w$ .

$2 w^{2} + 12 w + 18 + 3 w^{3} + 3 \cdot 6 (w \cdot w) + 3 w \cdot 9$

Multiply $w$ by $w$ .

$2 w^{2} + 12 w + 18 + 3 w^{3} + 3 \cdot 6 w^{2} + 3 w \cdot 9$

Multiply $3$ by $6$ .

$2 w^{2} + 12 w + 18 + 3 w^{3} + 18 w^{2} + 3 w \cdot 9$

Multiply $9$ by $3$ .

$2 w^{2} + 12 w + 18 + 3 w^{3} + 18 w^{2} + 27 w$

Simplify by adding terms.

Add $2 w^{2}$ and $18 w^{2}$ .

$20 w^{2} + 12 w + 18 + 3 w^{3} + 27 w$

Add $12 w$ and $27 w$ .

$20 w^{2} + 39 w + 18 + 3 w^{3}$