Trigonometry Examples

${(x + 3)}^{2} (x^{3} + 3 x^{2} + 3 x + 1) = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$(x + 3) (x + 3) (x^{3} + 3 x^{2} + 3 x + 1) = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$(x (x + 3) + 3 (x + 3)) (x^{3} + 3 x^{2} + 3 x + 1) = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 2.2

Apply the distributive property.

$(x \cdot x + x \cdot 3 + 3 (x + 3)) (x^{3} + 3 x^{2} + 3 x + 1) = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 2.3

Apply the distributive property.

$(x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) (x^{3} + 3 x^{2} + 3 x + 1) = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$(x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) (x^{3} + 3 x^{2} + 3 x + 1) = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 3.1.2

Move $3$ to the left of $x$ .

$(x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) (x^{3} + 3 x^{2} + 3 x + 1) = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 3.1.3

Multiply $3$ by $3$ .

$(x^{2} + 3 x + 3 x + 9) (x^{3} + 3 x^{2} + 3 x + 1) = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 3.2

Add $3 x$ and $3 x$ .

$(x^{2} + 6 x + 9) (x^{3} + 3 x^{2} + 3 x + 1) = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 4

Expand $(x^{2} + 6 x + 9) (x^{3} + 3 x^{2} + 3 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x^{3} + x^{2} (3 x^{2}) + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5

Simplify each term.

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Step 5.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 5.1.1

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

$x^{2 + 3} + x^{2} (3 x^{2}) + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.1.2

Add $2$ and $3$ .

$x^{5} + x^{2} (3 x^{2}) + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.2

Rewrite using the commutative property of multiplication.

$x^{5} + 3 x^{2} x^{2} + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.3

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

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Step 5.3.1

Move $x^{2}$ .

$x^{5} + 3 (x^{2} x^{2}) + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.3.2

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

$x^{5} + 3 x^{2 + 2} + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.3.3

Add $2$ and $2$ .

$x^{5} + 3 x^{4} + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.4

Rewrite using the commutative property of multiplication.

$x^{5} + 3 x^{4} + 3 x^{2} x + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.5

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

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Step 5.5.1

Move $x$ .

$x^{5} + 3 x^{4} + 3 (x \cdot x^{2}) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.5.2

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

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Step 5.5.2.1

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

$x^{5} + 3 x^{4} + 3 (x^{1} x^{2}) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.5.2.2

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

$x^{5} + 3 x^{4} + 3 x^{1 + 2} + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.5.3

Add $1$ and $2$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.6

Multiply $x^{2}$ by $1$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.7

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 5.7.1

Move $x^{3}$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 (x^{3} x) + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.7.2

Multiply $x^{3}$ by $x$ .

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Step 5.7.2.1

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

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 (x^{3} x^{1}) + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.7.2.2

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

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{3 + 1} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.7.3

Add $3$ and $1$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.8

Rewrite using the commutative property of multiplication.

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 \cdot 3 x \cdot x^{2} + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.9

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

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Step 5.9.1

Move $x^{2}$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 \cdot 3 (x^{2} x) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.9.2

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

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Step 5.9.2.1

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

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 \cdot 3 (x^{2} x^{1}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.9.2.2

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

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 \cdot 3 x^{2 + 1} + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.9.3

Add $2$ and $1$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 \cdot 3 x^{3} + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.10

Multiply $6$ by $3$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.11

Rewrite using the commutative property of multiplication.

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 6 \cdot 3 x \cdot x + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.12

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

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Step 5.12.1

Move $x$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 6 \cdot 3 (x \cdot x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.12.2

Multiply $x$ by $x$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 6 \cdot 3 x^{2} + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.13

Multiply $6$ by $3$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 18 x^{2} + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.14

Multiply $6$ by $1$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 18 x^{2} + 6 x + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.15

Multiply $3$ by $9$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 18 x^{2} + 6 x + 9 x^{3} + 27 x^{2} + 9 (3 x) + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.16

Multiply $3$ by $9$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 18 x^{2} + 6 x + 9 x^{3} + 27 x^{2} + 27 x + 9 \cdot 1 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 5.17

Multiply $9$ by $1$ .

$x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 18 x^{2} + 6 x + 9 x^{3} + 27 x^{2} + 27 x + 9 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 6

Add $3 x^{4}$ and $6 x^{4}$ .

$x^{5} + 9 x^{4} + 3 x^{3} + x^{2} + 18 x^{3} + 18 x^{2} + 6 x + 9 x^{3} + 27 x^{2} + 27 x + 9 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 7

Add $3 x^{3}$ and $18 x^{3}$ .

$x^{5} + 9 x^{4} + 21 x^{3} + x^{2} + 18 x^{2} + 6 x + 9 x^{3} + 27 x^{2} + 27 x + 9 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 8

Add $21 x^{3}$ and $9 x^{3}$ .

$x^{5} + 9 x^{4} + 30 x^{3} + x^{2} + 18 x^{2} + 6 x + 27 x^{2} + 27 x + 9 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 9

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

$x^{5} + 9 x^{4} + 30 x^{3} + 19 x^{2} + 6 x + 27 x^{2} + 27 x + 9 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 10

Add $19 x^{2}$ and $27 x^{2}$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 6 x + 27 x + 9 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 11

Add $6 x$ and $27 x$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = (x^{2} + 6 x + 9) {(x + 1)}^{3}$

Step 12

Use the Binomial Theorem.

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = (x^{2} + 6 x + 9) (x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3})$

Step 13

Simplify each term.

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Step 13.1

Multiply $3$ by $1$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = (x^{2} + 6 x + 9) (x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3})$

Step 13.2

One to any power is one.

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = (x^{2} + 6 x + 9) (x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3})$

Step 13.3

Multiply $3$ by $1$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = (x^{2} + 6 x + 9) (x^{3} + 3 x^{2} + 3 x + 1^{3})$

Step 13.4

One to any power is one.

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = (x^{2} + 6 x + 9) (x^{3} + 3 x^{2} + 3 x + 1)$

Step 14

Expand $(x^{2} + 6 x + 9) (x^{3} + 3 x^{2} + 3 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{2} x^{3} + x^{2} (3 x^{2}) + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15

Simplify each term.

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Step 15.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 15.1.1

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

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{2 + 3} + x^{2} (3 x^{2}) + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.1.2

Add $2$ and $3$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + x^{2} (3 x^{2}) + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.2

Rewrite using the commutative property of multiplication.

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{2} x^{2} + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.3

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

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Step 15.3.1

Move $x^{2}$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 (x^{2} x^{2}) + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.3.2

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

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{2 + 2} + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.3.3

Add $2$ and $2$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + x^{2} (3 x) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.4

Rewrite using the commutative property of multiplication.

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{2} x + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.5

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

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Step 15.5.1

Move $x$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 (x \cdot x^{2}) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.5.2

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

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Step 15.5.2.1

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

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 (x^{1} x^{2}) + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.5.2.2

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

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{1 + 2} + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.5.3

Add $1$ and $2$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} \cdot 1 + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.6

Multiply $x^{2}$ by $1$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x \cdot x^{3} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.7

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 15.7.1

Move $x^{3}$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 (x^{3} x) + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.7.2

Multiply $x^{3}$ by $x$ .

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Step 15.7.2.1

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

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 (x^{3} x^{1}) + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.7.2.2

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

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{3 + 1} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.7.3

Add $3$ and $1$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 x (3 x^{2}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.8

Rewrite using the commutative property of multiplication.

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 \cdot 3 x \cdot x^{2} + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.9

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

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Step 15.9.1

Move $x^{2}$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 \cdot 3 (x^{2} x) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.9.2

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

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Step 15.9.2.1

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

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 \cdot 3 (x^{2} x^{1}) + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.9.2.2

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

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 \cdot 3 x^{2 + 1} + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.9.3

Add $2$ and $1$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 6 \cdot 3 x^{3} + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.10

Multiply $6$ by $3$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 6 x (3 x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.11

Rewrite using the commutative property of multiplication.

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 6 \cdot 3 x \cdot x + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.12

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

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Step 15.12.1

Move $x$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 6 \cdot 3 (x \cdot x) + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.12.2

Multiply $x$ by $x$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 6 \cdot 3 x^{2} + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.13

Multiply $6$ by $3$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 18 x^{2} + 6 x \cdot 1 + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.14

Multiply $6$ by $1$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 18 x^{2} + 6 x + 9 x^{3} + 9 (3 x^{2}) + 9 (3 x) + 9 \cdot 1$

Step 15.15

Multiply $3$ by $9$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 18 x^{2} + 6 x + 9 x^{3} + 27 x^{2} + 9 (3 x) + 9 \cdot 1$

Step 15.16

Multiply $3$ by $9$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 18 x^{2} + 6 x + 9 x^{3} + 27 x^{2} + 27 x + 9 \cdot 1$

Step 15.17

Multiply $9$ by $1$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 3 x^{4} + 3 x^{3} + x^{2} + 6 x^{4} + 18 x^{3} + 18 x^{2} + 6 x + 9 x^{3} + 27 x^{2} + 27 x + 9$

Step 16

Add $3 x^{4}$ and $6 x^{4}$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 9 x^{4} + 3 x^{3} + x^{2} + 18 x^{3} + 18 x^{2} + 6 x + 9 x^{3} + 27 x^{2} + 27 x + 9$

Step 17

Add $3 x^{3}$ and $18 x^{3}$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 9 x^{4} + 21 x^{3} + x^{2} + 18 x^{2} + 6 x + 9 x^{3} + 27 x^{2} + 27 x + 9$

Step 18

Add $21 x^{3}$ and $9 x^{3}$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 9 x^{4} + 30 x^{3} + x^{2} + 18 x^{2} + 6 x + 27 x^{2} + 27 x + 9$

Step 19

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

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 9 x^{4} + 30 x^{3} + 19 x^{2} + 6 x + 27 x^{2} + 27 x + 9$

Step 20

Add $19 x^{2}$ and $27 x^{2}$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 6 x + 27 x + 9$

Step 21

Add $6 x$ and $27 x$ .

$x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9 = x^{5} + 9 x^{4} + 30 x^{3} + 46 x^{2} + 33 x + 9$

Step 22

Since the two sides have been shown to be equivalent, the equation is an identity.

${(x + 3)}^{2} (x^{3} + 3 x^{2} + 3 x + 1) = (x^{2} + 6 x + 9) {(x + 1)}^{3}$ is an identity.