Algebra Examples

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

Step 1

Set up the composite result function.

$g (h (x))$

Step 2

Evaluate $g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1)$ by substituting in the value of $h$ into $g$ .

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

Step 3

Simplify each term.

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Step 3.1

Simplify each term.

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Step 3.1.1

Use the Binomial Theorem.

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

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Raise $7$ to the power of $3$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 + 3 \cdot (7^{2} (- 3 x)) + 3 \cdot (7 {(- 3 x)}^{2}) + {(- 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1)$

Step 3.1.2.2

Raise $7$ to the power of $2$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 + 3 \cdot (49 (- 3 x)) + 3 \cdot (7 {(- 3 x)}^{2}) + {(- 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1)$

Step 3.1.2.3

Multiply $3$ by $49$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 + 147 (- 3 x) + 3 \cdot (7 {(- 3 x)}^{2}) + {(- 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1)$

Step 3.1.2.4

Multiply $- 3$ by $147$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 3 \cdot (7 {(- 3 x)}^{2}) + {(- 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1)$

Step 3.1.2.5

Multiply $3$ by $7$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 21 {(- 3 x)}^{2} + {(- 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1)$

Step 3.1.2.6

Apply the product rule to $- 3 x$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 21 ({(- 3)}^{2} x^{2}) + {(- 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1)$

Step 3.1.2.7

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

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 21 (9 x^{2}) + {(- 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1)$

Step 3.1.2.8

Multiply $9$ by $21$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} + {(- 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1)$

Step 3.1.2.9

Apply the product rule to $- 3 x$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} + {(- 3)}^{3} x^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1)$

Step 3.1.2.10

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

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1)$

Step 3.1.3

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

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 ((7 - 3 x) (7 - 3 x)) + 2 (7 - 3 x) - 1)$

Step 3.1.4

Expand $(7 - 3 x) (7 - 3 x)$ using the FOIL Method.

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Step 3.1.4.1

Apply the distributive property.

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 (7 (7 - 3 x) - 3 x (7 - 3 x)) + 2 (7 - 3 x) - 1)$

Step 3.1.4.2

Apply the distributive property.

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 (7 \cdot 7 + 7 (- 3 x) - 3 x (7 - 3 x)) + 2 (7 - 3 x) - 1)$

Step 3.1.4.3

Apply the distributive property.

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 (7 \cdot 7 + 7 (- 3 x) - 3 x \cdot 7 - 3 x (- 3 x)) + 2 (7 - 3 x) - 1)$

Step 3.1.5

Simplify and combine like terms.

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Step 3.1.5.1

Simplify each term.

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Step 3.1.5.1.1

Multiply $7$ by $7$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 (49 + 7 (- 3 x) - 3 x \cdot 7 - 3 x (- 3 x)) + 2 (7 - 3 x) - 1)$

Step 3.1.5.1.2

Multiply $- 3$ by $7$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 (49 - 21 x - 3 x \cdot 7 - 3 x (- 3 x)) + 2 (7 - 3 x) - 1)$

Step 3.1.5.1.3

Multiply $7$ by $- 3$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 (49 - 21 x - 21 x - 3 x (- 3 x)) + 2 (7 - 3 x) - 1)$

Step 3.1.5.1.4

Rewrite using the commutative property of multiplication.

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 (49 - 21 x - 21 x - 3 \cdot (- 3 x \cdot x)) + 2 (7 - 3 x) - 1)$

Step 3.1.5.1.5

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

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Step 3.1.5.1.5.1

Move $x$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 (49 - 21 x - 21 x - 3 \cdot (- 3 (x \cdot x))) + 2 (7 - 3 x) - 1)$

Step 3.1.5.1.5.2

Multiply $x$ by $x$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 (49 - 21 x - 21 x - 3 \cdot (- 3 x^{2})) + 2 (7 - 3 x) - 1)$

Step 3.1.5.1.6

Multiply $- 3$ by $- 3$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 (49 - 21 x - 21 x + 9 x^{2}) + 2 (7 - 3 x) - 1)$

Step 3.1.5.2

Subtract $21 x$ from $- 21 x$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 (49 - 42 x + 9 x^{2}) + 2 (7 - 3 x) - 1)$

Step 3.1.6

Apply the distributive property.

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 3 \cdot 49 - 3 (- 42 x) - 3 (9 x^{2}) + 2 (7 - 3 x) - 1)$

Step 3.1.7

Simplify.

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Step 3.1.7.1

Multiply $- 3$ by $49$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 147 - 3 (- 42 x) - 3 (9 x^{2}) + 2 (7 - 3 x) - 1)$

Step 3.1.7.2

Multiply $- 42$ by $- 3$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 147 + 126 x - 3 (9 x^{2}) + 2 (7 - 3 x) - 1)$

Step 3.1.7.3

Multiply $9$ by $- 3$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 147 + 126 x - 27 x^{2} + 2 (7 - 3 x) - 1)$

Step 3.1.8

Apply the distributive property.

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 147 + 126 x - 27 x^{2} + 2 \cdot 7 + 2 (- 3 x) - 1)$

Step 3.1.9

Multiply $2$ by $7$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 147 + 126 x - 27 x^{2} + 14 + 2 (- 3 x) - 1)$

Step 3.1.10

Multiply $- 3$ by $2$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (343 - 441 x + 189 x^{2} - 27 x^{3} - 147 + 126 x - 27 x^{2} + 14 - 6 x - 1)$

Step 3.2

Subtract $147$ from $343$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (- 441 x + 189 x^{2} - 27 x^{3} + 196 + 126 x - 27 x^{2} + 14 - 6 x - 1)$

Step 3.3

Add $- 441 x$ and $126 x$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (189 x^{2} - 27 x^{3} + 196 - 315 x - 27 x^{2} + 14 - 6 x - 1)$

Step 3.4

Subtract $27 x^{2}$ from $189 x^{2}$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (162 x^{2} - 27 x^{3} + 196 - 315 x + 14 - 6 x - 1)$

Step 3.5

Add $196$ and $14$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (162 x^{2} - 27 x^{3} - 315 x + 210 - 6 x - 1)$

Step 3.6

Subtract $6 x$ from $- 315 x$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (162 x^{2} - 27 x^{3} - 321 x + 210 - 1)$

Step 3.7

Subtract $1$ from $210$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (162 x^{2} - 27 x^{3} - 321 x + 209)$

Step 3.8

Apply the distributive property.

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 3 (162 x^{2}) - 3 (- 27 x^{3}) - 3 (- 321 x) - 3 \cdot 209$

Step 3.9

Simplify.

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Step 3.9.1

Multiply $162$ by $- 3$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 486 x^{2} - 3 (- 27 x^{3}) - 3 (- 321 x) - 3 \cdot 209$

Step 3.9.2

Multiply $- 27$ by $- 3$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 486 x^{2} + 81 x^{3} - 3 (- 321 x) - 3 \cdot 209$

Step 3.9.3

Multiply $- 321$ by $- 3$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 486 x^{2} + 81 x^{3} + 963 x - 3 \cdot 209$

Step 3.9.4

Multiply $- 3$ by $209$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 7 - 486 x^{2} + 81 x^{3} + 963 x - 627$

Step 4

Simplify the expression.

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Step 4.1

Subtract $627$ from $7$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = - 486 x^{2} + 81 x^{3} + 963 x - 620$

Step 4.2

Reorder $- 486 x^{2}$ and $81 x^{3}$ .

$g ({(7 - 3 x)}^{3} - 3 {(7 - 3 x)}^{2} + 2 (7 - 3 x) - 1) = 81 x^{3} - 486 x^{2} + 963 x - 620$