Algebra Examples

$f (x) = {(3 x^{2} - 5 x + 1)}^{3}$

Step 1

Use the Multinomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $3 x^{2}$ .

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

Step 2.1.2

Raise $3$ to the power of $3$ .

$f (x) = 27 {(x^{2})}^{3} + 3 {(3 x^{2})}^{2} (- 5 x) + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.3

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 2.1.3.1

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

$f (x) = 27 x^{2 \cdot 3} + 3 {(3 x^{2})}^{2} (- 5 x) + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.3.2

Multiply $2$ by $3$ .

$f (x) = 27 x^{6} + 3 {(3 x^{2})}^{2} (- 5 x) + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$f (x) = 27 x^{6} + 3 \cdot (- 5 {(3 x^{2})}^{2} x) + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.5

Multiply $3$ by $- 5$ .

$f (x) = 27 x^{6} - 15 {(3 x^{2})}^{2} x + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.6

Apply the product rule to $3 x^{2}$ .

$f (x) = 27 x^{6} - 15 (3^{2} {(x^{2})}^{2}) x + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.7

Raise $3$ to the power of $2$ .

$f (x) = 27 x^{6} - 15 (9 {(x^{2})}^{2}) x + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.8

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

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Step 2.1.8.1

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

$f (x) = 27 x^{6} - 15 (9 x^{2 \cdot 2}) x + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.8.2

Multiply $2$ by $2$ .

$f (x) = 27 x^{6} - 15 (9 x^{4}) x + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.9

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 2.1.9.1

Move $x$ .

$f (x) = 27 x^{6} - 15 (9 (x \cdot x^{4})) + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.9.2

Multiply $x$ by $x^{4}$ .

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Step 2.1.9.2.1

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

Step 2.1.9.2.2

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

$f (x) = 27 x^{6} - 15 (9 x^{1 + 4}) + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.9.3

Add $1$ and $4$ .

$f (x) = 27 x^{6} - 15 (9 x^{5}) + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.10

Multiply $9$ by $- 15$ .

$f (x) = 27 x^{6} - 135 x^{5} + 3 (3 x^{2}) {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.11

Multiply $3$ by $3$ .

$f (x) = 27 x^{6} - 135 x^{5} + 9 x^{2} {(- 5 x)}^{2} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.12

Apply the product rule to $- 5 x$ .

$f (x) = 27 x^{6} - 135 x^{5} + 9 x^{2} ({(- 5)}^{2} x^{2}) + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.13

Rewrite using the commutative property of multiplication.

$f (x) = 27 x^{6} - 135 x^{5} + 9 \cdot ({(- 5)}^{2} x^{2} x^{2}) + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.14

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

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Step 2.1.14.1

Move $x^{2}$ .

$f (x) = 27 x^{6} - 135 x^{5} + 9 \cdot ({(- 5)}^{2} (x^{2} x^{2})) + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.14.2

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

$f (x) = 27 x^{6} - 135 x^{5} + 9 \cdot ({(- 5)}^{2} x^{2 + 2}) + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.14.3

Add $2$ and $2$ .

$f (x) = 27 x^{6} - 135 x^{5} + 9 \cdot ({(- 5)}^{2} x^{4}) + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.15

Raise $- 5$ to the power of $2$ .

$f (x) = 27 x^{6} - 135 x^{5} + 9 \cdot (25 x^{4}) + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.16

Multiply $9$ by $25$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} + {(- 5 x)}^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.17

Apply the product rule to $- 5 x$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} + {(- 5)}^{3} x^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.18

Raise $- 5$ to the power of $3$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 3 {(3 x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.19

Apply the product rule to $3 x^{2}$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 3 (3^{2} {(x^{2})}^{2}) \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.20

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

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Step 2.1.20.1

Move $3^{2}$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 3^{2} \cdot (3 {(x^{2})}^{2}) \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.20.2

Multiply $3^{2}$ by $3$ .

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Step 2.1.20.2.1

Raise $3$ to the power of $1$ .

Step 2.1.20.2.2

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

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 3^{2 + 1} {(x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.20.3

Add $2$ and $1$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 3^{3} {(x^{2})}^{2} \cdot 1 + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.21

Simplify $3^{3} {(x^{2})}^{2} \cdot 1$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 3^{3} {(x^{2})}^{2} + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.22

Raise $3$ to the power of $3$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 {(x^{2})}^{2} + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.23

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

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Step 2.1.23.1

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

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{2 \cdot 2} + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.23.2

Multiply $2$ by $2$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} + 6 (3 x^{2}) (- 5 x) \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.24

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

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Step 2.1.24.1

Move $x$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} + 6 (3 (x \cdot x^{2})) \cdot - 5 \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.24.2

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

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Step 2.1.24.2.1

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

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} + 6 (3 (x \cdot x^{2})) \cdot - 5 \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.24.2.2

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

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} + 6 (3 x^{1 + 2}) \cdot - 5 \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.24.3

Add $1$ and $2$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} + 6 (3 x^{3}) \cdot - 5 \cdot 1 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.25

Simplify $6 (3 x^{3}) \cdot - 5 \cdot 1$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} + 6 (3 x^{3}) \cdot - 5 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.26

Multiply $3$ by $6$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} + 18 x^{3} \cdot - 5 + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.27

Multiply $- 5$ by $18$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 3 {(- 5 x)}^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.28

Apply the product rule to $- 5 x$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 3 ({(- 5)}^{2} x^{2}) \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.29

Raise $- 5$ to the power of $2$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 3 (25 x^{2}) \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.30

Multiply $25$ by $3$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 75 x^{2} \cdot 1 + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.31

Multiply $75$ by $1$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 75 x^{2} + 3 (3 x^{2}) \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.32

Multiply $3$ by $3$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 75 x^{2} + 9 x^{2} \cdot 1^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.33

One to any power is one.

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 75 x^{2} + 9 x^{2} \cdot 1 + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.34

Multiply $9$ by $1$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 75 x^{2} + 9 x^{2} + 3 (- 5 x) \cdot 1^{2} + 1^{3}$

Step 2.1.35

Multiply $- 5$ by $3$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 75 x^{2} + 9 x^{2} - 15 x \cdot 1^{2} + 1^{3}$

Step 2.1.36

One to any power is one.

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 75 x^{2} + 9 x^{2} - 15 x \cdot 1 + 1^{3}$

Step 2.1.37

Multiply $- 15$ by $1$ .

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 75 x^{2} + 9 x^{2} - 15 x + 1^{3}$

Step 2.1.38

One to any power is one.

$f (x) = 27 x^{6} - 135 x^{5} + 225 x^{4} - 125 x^{3} + 27 x^{4} - 90 x^{3} + 75 x^{2} + 9 x^{2} - 15 x + 1$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $225 x^{4}$ and $27 x^{4}$ .

$f (x) = 27 x^{6} - 135 x^{5} + 252 x^{4} - 125 x^{3} - 90 x^{3} + 75 x^{2} + 9 x^{2} - 15 x + 1$

Step 2.2.2

Subtract $90 x^{3}$ from $- 125 x^{3}$ .

$f (x) = 27 x^{6} - 135 x^{5} + 252 x^{4} - 215 x^{3} + 75 x^{2} + 9 x^{2} - 15 x + 1$

Step 2.2.3

Add $75 x^{2}$ and $9 x^{2}$ .

$f (x) = 27 x^{6} - 135 x^{5} + 252 x^{4} - 215 x^{3} + 84 x^{2} - 15 x + 1$

Step 3