Algebra Examples

$f (x) = 3 x^{3} - 2 x^{2} + 7 x - 5$ , $g (x) = 5 x^{2} + x - 1$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (3 x^{3} - 2 x^{2} + 7 x - 5)$ by substituting in the value of $f$ into $g$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 {(3 x^{3} - 2 x^{2} + 7 x - 5)}^{2} + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 3

Remove parentheses.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 {(3 x^{3} - 2 x^{2} + 7 x - 5)}^{2} + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4

Simplify each term.

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Step 4.1

Rewrite ${(3 x^{3} - 2 x^{2} + 7 x - 5)}^{2}$ as $(3 x^{3} - 2 x^{2} + 7 x - 5) (3 x^{3} - 2 x^{2} + 7 x - 5)$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 ((3 x^{3} - 2 x^{2} + 7 x - 5) (3 x^{3} - 2 x^{2} + 7 x - 5)) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.2

Expand $(3 x^{3} - 2 x^{2} + 7 x - 5) (3 x^{3} - 2 x^{2} + 7 x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (3 x^{3} (3 x^{3}) + 3 x^{3} (- 2 x^{2}) + 3 x^{3} (7 x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3

Simplify each term.

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Step 4.3.1

Rewrite using the commutative property of multiplication.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (3 \cdot (3 x^{3} x^{3}) + 3 x^{3} (- 2 x^{2}) + 3 x^{3} (7 x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.2

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

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Step 4.3.2.1

Move $x^{3}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (3 \cdot (3 (x^{3} x^{3})) + 3 x^{3} (- 2 x^{2}) + 3 x^{3} (7 x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.2.2

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

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (3 \cdot (3 x^{3 + 3}) + 3 x^{3} (- 2 x^{2}) + 3 x^{3} (7 x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.2.3

Add $3$ and $3$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (3 \cdot (3 x^{6}) + 3 x^{3} (- 2 x^{2}) + 3 x^{3} (7 x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.3

Multiply $3$ by $3$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} + 3 x^{3} (- 2 x^{2}) + 3 x^{3} (7 x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.4

Rewrite using the commutative property of multiplication.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} + 3 \cdot (- 2 x^{3} x^{2}) + 3 x^{3} (7 x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.5

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

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Step 4.3.5.1

Move $x^{2}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} + 3 \cdot (- 2 (x^{2} x^{3})) + 3 x^{3} (7 x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.5.2

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

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} + 3 \cdot (- 2 x^{2 + 3}) + 3 x^{3} (7 x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.5.3

Add $2$ and $3$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} + 3 \cdot (- 2 x^{5}) + 3 x^{3} (7 x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.6

Multiply $3$ by $- 2$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 3 x^{3} (7 x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.7

Rewrite using the commutative property of multiplication.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 3 \cdot (7 x^{3} x) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.8

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

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Step 4.3.8.1

Move $x$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 3 \cdot (7 (x \cdot x^{3})) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.8.2

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

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Step 4.3.8.2.1

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

Step 4.3.8.2.2

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

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 3 \cdot (7 x^{1 + 3}) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.8.3

Add $1$ and $3$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 3 \cdot (7 x^{4}) + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.9

Multiply $3$ by $7$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} + 3 x^{3} \cdot - 5 - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.10

Multiply $- 5$ by $3$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 2 x^{2} (3 x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.11

Rewrite using the commutative property of multiplication.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 2 \cdot (3 x^{2} x^{3}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.12

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

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Step 4.3.12.1

Move $x^{3}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 2 \cdot (3 (x^{3} x^{2})) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.12.2

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

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 2 \cdot (3 x^{3 + 2}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.12.3

Add $3$ and $2$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 2 \cdot (3 x^{5}) - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.13

Multiply $- 2$ by $3$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} - 2 x^{2} (- 2 x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.14

Rewrite using the commutative property of multiplication.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} - 2 \cdot (- 2 x^{2} x^{2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.15

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

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Step 4.3.15.1

Move $x^{2}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} - 2 \cdot (- 2 (x^{2} x^{2})) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.15.2

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

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} - 2 \cdot (- 2 x^{2 + 2}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.15.3

Add $2$ and $2$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} - 2 \cdot (- 2 x^{4}) - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.16

Multiply $- 2$ by $- 2$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 2 x^{2} (7 x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.17

Rewrite using the commutative property of multiplication.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 2 \cdot (7 x^{2} x) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.18

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

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Step 4.3.18.1

Move $x$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 2 \cdot (7 (x \cdot x^{2})) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.18.2

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

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Step 4.3.18.2.1

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

Step 4.3.18.2.2

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

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 2 \cdot (7 x^{1 + 2}) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.18.3

Add $1$ and $2$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 2 \cdot (7 x^{3}) - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.19

Multiply $- 2$ by $7$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} - 2 x^{2} \cdot - 5 + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.20

Multiply $- 5$ by $- 2$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 7 x (3 x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.21

Rewrite using the commutative property of multiplication.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 7 \cdot (3 x \cdot x^{3}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.22

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

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Step 4.3.22.1

Move $x^{3}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 7 \cdot (3 (x^{3} x)) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.22.2

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

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Step 4.3.22.2.1

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

Step 4.3.22.2.2

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

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 7 \cdot (3 x^{3 + 1}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.22.3

Add $3$ and $1$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 7 \cdot (3 x^{4}) + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.23

Multiply $7$ by $3$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} + 7 x (- 2 x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.24

Rewrite using the commutative property of multiplication.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} + 7 \cdot (- 2 x \cdot x^{2}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.25

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

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Step 4.3.25.1

Move $x^{2}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} + 7 \cdot (- 2 (x^{2} x)) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.25.2

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

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Step 4.3.25.2.1

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

Step 4.3.25.2.2

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

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} + 7 \cdot (- 2 x^{2 + 1}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.25.3

Add $2$ and $1$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} + 7 \cdot (- 2 x^{3}) + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.26

Multiply $7$ by $- 2$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 7 x (7 x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.27

Rewrite using the commutative property of multiplication.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 7 \cdot (7 x \cdot x) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.28

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

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Step 4.3.28.1

Move $x$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 7 \cdot (7 (x \cdot x)) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.28.2

Multiply $x$ by $x$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 7 \cdot (7 x^{2}) + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.29

Multiply $7$ by $7$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 49 x^{2} + 7 x \cdot - 5 - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.30

Multiply $- 5$ by $7$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 49 x^{2} - 35 x - 5 (3 x^{3}) - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.31

Multiply $3$ by $- 5$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 49 x^{2} - 35 x - 15 x^{3} - 5 (- 2 x^{2}) - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.32

Multiply $- 2$ by $- 5$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 49 x^{2} - 35 x - 15 x^{3} + 10 x^{2} - 5 (7 x) - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.33

Multiply $7$ by $- 5$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 49 x^{2} - 35 x - 15 x^{3} + 10 x^{2} - 35 x - 5 \cdot - 5) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.3.34

Multiply $- 5$ by $- 5$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 6 x^{5} + 21 x^{4} - 15 x^{3} - 6 x^{5} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 49 x^{2} - 35 x - 15 x^{3} + 10 x^{2} - 35 x + 25) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.4

Subtract $6 x^{5}$ from $- 6 x^{5}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 12 x^{5} + 21 x^{4} - 15 x^{3} + 4 x^{4} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 49 x^{2} - 35 x - 15 x^{3} + 10 x^{2} - 35 x + 25) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.5

Add $21 x^{4}$ and $4 x^{4}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 12 x^{5} + 25 x^{4} - 15 x^{3} - 14 x^{3} + 10 x^{2} + 21 x^{4} - 14 x^{3} + 49 x^{2} - 35 x - 15 x^{3} + 10 x^{2} - 35 x + 25) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.6

Add $25 x^{4}$ and $21 x^{4}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 12 x^{5} + 46 x^{4} - 15 x^{3} - 14 x^{3} + 10 x^{2} - 14 x^{3} + 49 x^{2} - 35 x - 15 x^{3} + 10 x^{2} - 35 x + 25) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.7

Subtract $14 x^{3}$ from $- 15 x^{3}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 12 x^{5} + 46 x^{4} - 29 x^{3} + 10 x^{2} - 14 x^{3} + 49 x^{2} - 35 x - 15 x^{3} + 10 x^{2} - 35 x + 25) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.8

Subtract $14 x^{3}$ from $- 29 x^{3}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 12 x^{5} + 46 x^{4} - 43 x^{3} + 10 x^{2} + 49 x^{2} - 35 x - 15 x^{3} + 10 x^{2} - 35 x + 25) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.9

Subtract $15 x^{3}$ from $- 43 x^{3}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 12 x^{5} + 46 x^{4} - 58 x^{3} + 10 x^{2} + 49 x^{2} - 35 x + 10 x^{2} - 35 x + 25) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.10

Add $10 x^{2}$ and $49 x^{2}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 12 x^{5} + 46 x^{4} - 58 x^{3} + 59 x^{2} - 35 x + 10 x^{2} - 35 x + 25) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.11

Add $59 x^{2}$ and $10 x^{2}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 12 x^{5} + 46 x^{4} - 58 x^{3} + 69 x^{2} - 35 x - 35 x + 25) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.12

Subtract $35 x$ from $- 35 x$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6} - 12 x^{5} + 46 x^{4} - 58 x^{3} + 69 x^{2} - 70 x + 25) + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.13

Apply the distributive property.

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 5 (9 x^{6}) + 5 (- 12 x^{5}) + 5 (46 x^{4}) + 5 (- 58 x^{3}) + 5 (69 x^{2}) + 5 (- 70 x) + 5 \cdot 25 + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.14

Simplify.

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Step 4.14.1

Multiply $9$ by $5$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} + 5 (- 12 x^{5}) + 5 (46 x^{4}) + 5 (- 58 x^{3}) + 5 (69 x^{2}) + 5 (- 70 x) + 5 \cdot 25 + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.14.2

Multiply $- 12$ by $5$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} - 60 x^{5} + 5 (46 x^{4}) + 5 (- 58 x^{3}) + 5 (69 x^{2}) + 5 (- 70 x) + 5 \cdot 25 + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.14.3

Multiply $46$ by $5$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} - 60 x^{5} + 230 x^{4} + 5 (- 58 x^{3}) + 5 (69 x^{2}) + 5 (- 70 x) + 5 \cdot 25 + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.14.4

Multiply $- 58$ by $5$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} - 60 x^{5} + 230 x^{4} - 290 x^{3} + 5 (69 x^{2}) + 5 (- 70 x) + 5 \cdot 25 + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.14.5

Multiply $69$ by $5$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} - 60 x^{5} + 230 x^{4} - 290 x^{3} + 345 x^{2} + 5 (- 70 x) + 5 \cdot 25 + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.14.6

Multiply $- 70$ by $5$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} - 60 x^{5} + 230 x^{4} - 290 x^{3} + 345 x^{2} - 350 x + 5 \cdot 25 + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 4.14.7

Multiply $5$ by $25$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} - 60 x^{5} + 230 x^{4} - 290 x^{3} + 345 x^{2} - 350 x + 125 + 3 x^{3} - 2 x^{2} + 7 x - 5 - 1$

Step 5

Simplify by adding terms.

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Step 5.1

Add $- 290 x^{3}$ and $3 x^{3}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} - 60 x^{5} + 230 x^{4} - 287 x^{3} + 345 x^{2} - 350 x + 125 - 2 x^{2} + 7 x - 5 - 1$

Step 5.2

Subtract $2 x^{2}$ from $345 x^{2}$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} - 60 x^{5} + 230 x^{4} - 287 x^{3} + 343 x^{2} - 350 x + 125 + 7 x - 5 - 1$

Step 5.3

Add $- 350 x$ and $7 x$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} - 60 x^{5} + 230 x^{4} - 287 x^{3} + 343 x^{2} - 343 x + 125 - 5 - 1$

Step 5.4

Simplify by subtracting numbers.

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Step 5.4.1

Subtract $5$ from $125$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} - 60 x^{5} + 230 x^{4} - 287 x^{3} + 343 x^{2} - 343 x + 120 - 1$

Step 5.4.2

Subtract $1$ from $120$ .

$g (3 x^{3} - 2 x^{2} + 7 x - 5) = 45 x^{6} - 60 x^{5} + 230 x^{4} - 287 x^{3} + 343 x^{2} - 343 x + 119$