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.

$h (g (x))$

Step 2

Evaluate $h (7 - 3 x)$ by substituting in the value of $g$ into $h$ .

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

Step 3

Simplify each term.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Multiply $- 3$ by $7$ .

$h (7 - 3 x) = {(7 - 21 - 3 (- 3 x))}^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.1.3

Multiply $- 3$ by $- 3$ .

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

Step 3.2

Subtract $21$ from $7$ .

$h (7 - 3 x) = {(- 14 + 9 x)}^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.3

Use the Binomial Theorem.

$h (7 - 3 x) = {(- 14)}^{3} + 3 {(- 14)}^{2} (9 x) + 3 \cdot (- 14 {(9 x)}^{2}) + {(9 x)}^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.4

Simplify each term.

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Step 3.4.1

Raise $- 14$ to the power of $3$ .

$h (7 - 3 x) = - 2744 + 3 {(- 14)}^{2} (9 x) + 3 \cdot (- 14 {(9 x)}^{2}) + {(9 x)}^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.4.2

Raise $- 14$ to the power of $2$ .

$h (7 - 3 x) = - 2744 + 3 \cdot (196 (9 x)) + 3 \cdot (- 14 {(9 x)}^{2}) + {(9 x)}^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.4.3

Multiply $3$ by $196$ .

$h (7 - 3 x) = - 2744 + 588 (9 x) + 3 \cdot (- 14 {(9 x)}^{2}) + {(9 x)}^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.4.4

Multiply $9$ by $588$ .

$h (7 - 3 x) = - 2744 + 5292 x + 3 \cdot (- 14 {(9 x)}^{2}) + {(9 x)}^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.4.5

Multiply $3$ by $- 14$ .

$h (7 - 3 x) = - 2744 + 5292 x - 42 {(9 x)}^{2} + {(9 x)}^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.4.6

Apply the product rule to $9 x$ .

$h (7 - 3 x) = - 2744 + 5292 x - 42 (9^{2} x^{2}) + {(9 x)}^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.4.7

Raise $9$ to the power of $2$ .

$h (7 - 3 x) = - 2744 + 5292 x - 42 (81 x^{2}) + {(9 x)}^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.4.8

Multiply $81$ by $- 42$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + {(9 x)}^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.4.9

Apply the product rule to $9 x$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 9^{3} x^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.4.10

Raise $9$ to the power of $3$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 {(7 - 3 (7 - 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.5

Simplify each term.

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Step 3.5.1

Apply the distributive property.

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 {(7 - 3 \cdot 7 - 3 (- 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.5.2

Multiply $- 3$ by $7$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 {(7 - 21 - 3 (- 3 x))}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.5.3

Multiply $- 3$ by $- 3$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 {(7 - 21 + 9 x)}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.6

Subtract $21$ from $7$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 {(- 14 + 9 x)}^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.7

Rewrite ${(- 14 + 9 x)}^{2}$ as $(- 14 + 9 x) (- 14 + 9 x)$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 ((- 14 + 9 x) (- 14 + 9 x)) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.8

Expand $(- 14 + 9 x) (- 14 + 9 x)$ using the FOIL Method.

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Step 3.8.1

Apply the distributive property.

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 (- 14 (- 14 + 9 x) + 9 x (- 14 + 9 x)) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.8.2

Apply the distributive property.

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 (- 14 \cdot - 14 - 14 (9 x) + 9 x (- 14 + 9 x)) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.8.3

Apply the distributive property.

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 (- 14 \cdot - 14 - 14 (9 x) + 9 x \cdot - 14 + 9 x (9 x)) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.9

Simplify and combine like terms.

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Step 3.9.1

Simplify each term.

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Step 3.9.1.1

Multiply $- 14$ by $- 14$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 (196 - 14 (9 x) + 9 x \cdot - 14 + 9 x (9 x)) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.9.1.2

Multiply $9$ by $- 14$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 (196 - 126 x + 9 x \cdot - 14 + 9 x (9 x)) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.9.1.3

Multiply $- 14$ by $9$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 (196 - 126 x - 126 x + 9 x (9 x)) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.9.1.4

Rewrite using the commutative property of multiplication.

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 (196 - 126 x - 126 x + 9 \cdot (9 x \cdot x)) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.9.1.5

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

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Step 3.9.1.5.1

Move $x$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 (196 - 126 x - 126 x + 9 \cdot (9 (x \cdot x))) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.9.1.5.2

Multiply $x$ by $x$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 (196 - 126 x - 126 x + 9 \cdot (9 x^{2})) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.9.1.6

Multiply $9$ by $9$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 (196 - 126 x - 126 x + 81 x^{2}) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.9.2

Subtract $126 x$ from $- 126 x$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 (196 - 252 x + 81 x^{2}) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.10

Apply the distributive property.

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 3 \cdot 196 - 3 (- 252 x) - 3 (81 x^{2}) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.11

Simplify.

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Step 3.11.1

Multiply $- 3$ by $196$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 588 - 3 (- 252 x) - 3 (81 x^{2}) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.11.2

Multiply $- 252$ by $- 3$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 588 + 756 x - 3 (81 x^{2}) + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.11.3

Multiply $81$ by $- 3$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 588 + 756 x - 243 x^{2} + 2 (7 - 3 (7 - 3 x)) - 1$

Step 3.12

Simplify each term.

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Step 3.12.1

Apply the distributive property.

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 588 + 756 x - 243 x^{2} + 2 (7 - 3 \cdot 7 - 3 (- 3 x)) - 1$

Step 3.12.2

Multiply $- 3$ by $7$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 588 + 756 x - 243 x^{2} + 2 (7 - 21 - 3 (- 3 x)) - 1$

Step 3.12.3

Multiply $- 3$ by $- 3$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 588 + 756 x - 243 x^{2} + 2 (7 - 21 + 9 x) - 1$

Step 3.13

Subtract $21$ from $7$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 588 + 756 x - 243 x^{2} + 2 (- 14 + 9 x) - 1$

Step 3.14

Apply the distributive property.

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 588 + 756 x - 243 x^{2} + 2 \cdot - 14 + 2 (9 x) - 1$

Step 3.15

Multiply $2$ by $- 14$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 588 + 756 x - 243 x^{2} - 28 + 2 (9 x) - 1$

Step 3.16

Multiply $9$ by $2$ .

$h (7 - 3 x) = - 2744 + 5292 x - 3402 x^{2} + 729 x^{3} - 588 + 756 x - 243 x^{2} - 28 + 18 x - 1$

Step 4

Simplify by adding terms.

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Step 4.1

Subtract $588$ from $- 2744$ .

$h (7 - 3 x) = 5292 x - 3402 x^{2} + 729 x^{3} - 3332 + 756 x - 243 x^{2} - 28 + 18 x - 1$

Step 4.2

Add $5292 x$ and $756 x$ .

$h (7 - 3 x) = - 3402 x^{2} + 729 x^{3} - 3332 + 6048 x - 243 x^{2} - 28 + 18 x - 1$

Step 4.3

Subtract $243 x^{2}$ from $- 3402 x^{2}$ .

$h (7 - 3 x) = - 3645 x^{2} + 729 x^{3} - 3332 + 6048 x - 28 + 18 x - 1$

Step 4.4

Subtract $28$ from $- 3332$ .

$h (7 - 3 x) = - 3645 x^{2} + 729 x^{3} + 6048 x - 3360 + 18 x - 1$

Step 4.5

Add $6048 x$ and $18 x$ .

$h (7 - 3 x) = - 3645 x^{2} + 729 x^{3} + 6066 x - 3360 - 1$

Step 4.6

Simplify the expression.

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Step 4.6.1

Subtract $1$ from $- 3360$ .

$h (7 - 3 x) = - 3645 x^{2} + 729 x^{3} + 6066 x - 3361$

Step 4.6.2

Reorder $- 3645 x^{2}$ and $729 x^{3}$ .

$h (7 - 3 x) = 729 x^{3} - 3645 x^{2} + 6066 x - 3361$