Algebra Examples

$f (x) = x^{3}$ , $g (x) = x^{2} + 4 x - 6$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x^{2} + 4 x - 6)$ by substituting in the value of $g$ into $f$ .

$f (x^{2} + 4 x - 6) = {(x^{2} + 4 x - 6)}^{3}$

Step 3

Use the Multinomial Theorem.

$f (x^{2} + 4 x - 6) = {(x^{2})}^{3} + 3 {(x^{2})}^{2} (4 x) + 3 x^{2} {(4 x)}^{2} + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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Step 4.1.1.1

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

$f (x^{2} + 4 x - 6) = x^{2 \cdot 3} + 3 {(x^{2})}^{2} (4 x) + 3 x^{2} {(4 x)}^{2} + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.1.2

Multiply $2$ by $3$ .

$f (x^{2} + 4 x - 6) = x^{6} + 3 {(x^{2})}^{2} (4 x) + 3 x^{2} {(4 x)}^{2} + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.2

Rewrite using the commutative property of multiplication.

$f (x^{2} + 4 x - 6) = x^{6} + 3 \cdot (4 {(x^{2})}^{2} x) + 3 x^{2} {(4 x)}^{2} + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.3

Multiply $3$ by $4$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 {(x^{2})}^{2} x + 3 x^{2} {(4 x)}^{2} + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.4

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

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Step 4.1.4.1

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

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{2 \cdot 2} x + 3 x^{2} {(4 x)}^{2} + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.4.2

Multiply $2$ by $2$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{4} x + 3 x^{2} {(4 x)}^{2} + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.5

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

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Step 4.1.5.1

Move $x$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 (x \cdot x^{4}) + 3 x^{2} {(4 x)}^{2} + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.5.2

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

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Step 4.1.5.2.1

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

Step 4.1.5.2.2

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

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{1 + 4} + 3 x^{2} {(4 x)}^{2} + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.5.3

Add $1$ and $4$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 3 x^{2} {(4 x)}^{2} + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.6

Apply the product rule to $4 x$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 3 x^{2} (4^{2} x^{2}) + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.7

Rewrite using the commutative property of multiplication.

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 3 \cdot (4^{2} x^{2} x^{2}) + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.8

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

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Step 4.1.8.1

Move $x^{2}$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 3 \cdot (4^{2} (x^{2} x^{2})) + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.8.2

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

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 3 \cdot (4^{2} x^{2 + 2}) + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.8.3

Add $2$ and $2$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 3 \cdot (4^{2} x^{4}) + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.9

Raise $4$ to the power of $2$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 3 \cdot (16 x^{4}) + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.10

Multiply $3$ by $16$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + {(4 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.11

Apply the product rule to $4 x$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 4^{3} x^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.12

Raise $4$ to the power of $3$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} + 3 {(x^{2})}^{2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.13

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

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Step 4.1.13.1

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

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} + 3 x^{2 \cdot 2} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.13.2

Multiply $2$ by $2$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} + 3 x^{4} \cdot - 6 + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.14

Multiply $- 6$ by $3$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} + 6 x^{2} (4 x) \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.15

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

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Step 4.1.15.1

Move $x$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} + 6 (x \cdot x^{2}) \cdot 4 \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.15.2

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

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Step 4.1.15.2.1

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

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} + 6 (x \cdot x^{2}) \cdot 4 \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.15.2.2

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

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} + 6 x^{1 + 2} \cdot 4 \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.15.3

Add $1$ and $2$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} + 6 x^{3} \cdot 4 \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.16

Multiply $4$ by $6$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} + 24 x^{3} \cdot - 6 + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.17

Multiply $- 6$ by $24$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} - 144 x^{3} + 3 {(4 x)}^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.18

Apply the product rule to $4 x$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} - 144 x^{3} + 3 (4^{2} x^{2}) \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.19

Raise $4$ to the power of $2$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} - 144 x^{3} + 3 (16 x^{2}) \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.20

Multiply $16$ by $3$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} - 144 x^{3} + 48 x^{2} \cdot - 6 + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.21

Multiply $- 6$ by $48$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} - 144 x^{3} - 288 x^{2} + 3 x^{2} {(- 6)}^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.22

Raise $- 6$ to the power of $2$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} - 144 x^{3} - 288 x^{2} + 3 x^{2} \cdot 36 + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.23

Multiply $36$ by $3$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} - 144 x^{3} - 288 x^{2} + 108 x^{2} + 3 (4 x) {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.24

Multiply $4$ by $3$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} - 144 x^{3} - 288 x^{2} + 108 x^{2} + 12 x {(- 6)}^{2} + {(- 6)}^{3}$

Step 4.1.25

Raise $- 6$ to the power of $2$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} - 144 x^{3} - 288 x^{2} + 108 x^{2} + 12 x \cdot 36 + {(- 6)}^{3}$

Step 4.1.26

Multiply $36$ by $12$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} - 144 x^{3} - 288 x^{2} + 108 x^{2} + 432 x + {(- 6)}^{3}$

Step 4.1.27

Raise $- 6$ to the power of $3$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 48 x^{4} + 64 x^{3} - 18 x^{4} - 144 x^{3} - 288 x^{2} + 108 x^{2} + 432 x - 216$

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Subtract $18 x^{4}$ from $48 x^{4}$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 30 x^{4} + 64 x^{3} - 144 x^{3} - 288 x^{2} + 108 x^{2} + 432 x - 216$

Step 4.2.2

Subtract $144 x^{3}$ from $64 x^{3}$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 30 x^{4} - 80 x^{3} - 288 x^{2} + 108 x^{2} + 432 x - 216$

Step 4.2.3

Add $- 288 x^{2}$ and $108 x^{2}$ .

$f (x^{2} + 4 x - 6) = x^{6} + 12 x^{5} + 30 x^{4} - 80 x^{3} - 180 x^{2} + 432 x - 216$