Precalculus Examples

$5 {(x^{6} + 1)}^{4} (6 x^{5}) {(3 x + 2)}^{3} + 3 {(3 x + 2)}^{2} (3) {(x^{6} + 1)}^{5}$

Step 1

Multiply $3 {(3 x + 2)}^{2}$ by $3$ .

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

Step 2

Factor ${(x^{6} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2}$ out of $5 {(x^{6} + 1)}^{4} (6 x^{5}) {(3 x + 2)}^{3} + 3 {(3 x + 2)}^{2} \cdot 3 {(x^{6} + 1)}^{5}$ .

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Step 2.1

Factor ${(x^{6} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2}$ out of $5 {(x^{6} + 1)}^{4} (6 x^{5}) {(3 x + 2)}^{3}$ .

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

Step 2.2

Factor ${(x^{6} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2}$ out of $3 {(3 x + 2)}^{2} \cdot 3 {(x^{6} + 1)}^{5}$ .

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

Step 2.3

Factor ${(x^{6} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2}$ out of ${(x^{6} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (5 (2 x^{5}) (3 x + 2)) + {(x^{6} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (3 (x^{6} + 1))$ .

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

Step 3

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

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

Step 4

Rewrite $1$ as $1^{3}$ .

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

Step 5

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x^{2}$ and $b = 1$ .

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

Step 6

Simplify.

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Step 6.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 6.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 6.1.2

Multiply $2$ by $2$ .

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

Step 6.2

Multiply $- 1$ by $1$ .

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

Step 6.3

One to any power is one.

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

Step 7

Apply the product rule to $(x^{2} + 1) (x^{4} - x^{2} + 1)$ .

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

Step 8

Multiply $2$ by $5$ .

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (10 x^{5} (3 x + 2) + 3 (x^{6} + 1))$

Step 9

Apply the distributive property.

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (10 x^{5} (3 x) + 10 x^{5} \cdot 2 + 3 (x^{6} + 1))$

Step 10

Rewrite using the commutative property of multiplication.

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (10 \cdot 3 x^{5} x + 10 x^{5} \cdot 2 + 3 (x^{6} + 1))$

Step 11

Multiply $2$ by $10$ .

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (10 \cdot 3 x^{5} x + 20 x^{5} + 3 (x^{6} + 1))$

Step 12

Simplify each term.

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Step 12.1

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 12.1.1

Move $x$ .

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (10 \cdot 3 (x \cdot x^{5}) + 20 x^{5} + 3 (x^{6} + 1))$

Step 12.1.2

Multiply $x$ by $x^{5}$ .

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Step 12.1.2.1

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

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (10 \cdot 3 (x^{1} x^{5}) + 20 x^{5} + 3 (x^{6} + 1))$

Step 12.1.2.2

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

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (10 \cdot 3 x^{1 + 5} + 20 x^{5} + 3 (x^{6} + 1))$

Step 12.1.3

Add $1$ and $5$ .

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (10 \cdot 3 x^{6} + 20 x^{5} + 3 (x^{6} + 1))$

Step 12.2

Multiply $10$ by $3$ .

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (30 x^{6} + 20 x^{5} + 3 (x^{6} + 1))$

Step 13

Apply the distributive property.

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (30 x^{6} + 20 x^{5} + 3 x^{6} + 3 \cdot 1)$

Step 14

Multiply $3$ by $1$ .

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (30 x^{6} + 20 x^{5} + 3 x^{6} + 3)$

Step 15

Add $30 x^{6}$ and $3 x^{6}$ .

${(x^{2} + 1)}^{4} {(x^{4} - x^{2} + 1)}^{4} \cdot 3 {(3 x + 2)}^{2} (33 x^{6} + 20 x^{5} + 3)$