Algebra Examples

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

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

Step 3

Use the Binomial Theorem.

$f (- 2 - 3 x^{2}) = {(- 2)}^{5} + 5 {(- 2)}^{4} (- 3 x^{2}) + 10 {(- 2)}^{3} {(- 3 x^{2})}^{2} + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

$f (- 2 - 3 x^{2}) = - 32 + 5 {(- 2)}^{4} (- 3 x^{2}) + 10 {(- 2)}^{3} {(- 3 x^{2})}^{2} + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.2

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

$f (- 2 - 3 x^{2}) = - 32 + 5 \cdot (16 (- 3 x^{2})) + 10 {(- 2)}^{3} {(- 3 x^{2})}^{2} + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.3

Multiply $5$ by $16$ .

$f (- 2 - 3 x^{2}) = - 32 + 80 (- 3 x^{2}) + 10 {(- 2)}^{3} {(- 3 x^{2})}^{2} + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.4

Multiply $- 3$ by $80$ .

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} + 10 {(- 2)}^{3} {(- 3 x^{2})}^{2} + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.5

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} + 10 \cdot (- 8 {(- 3 x^{2})}^{2}) + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.6

Multiply $10$ by $- 8$ .

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 80 {(- 3 x^{2})}^{2} + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.7

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 80 ({(- 3)}^{2} {(x^{2})}^{2}) + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.8

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 80 (9 {(x^{2})}^{2}) + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.9

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

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Step 4.1.9.1

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 80 (9 x^{2 \cdot 2}) + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.9.2

Multiply $2$ by $2$ .

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 80 (9 x^{4}) + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.10

Multiply $9$ by $- 80$ .

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} + 10 {(- 2)}^{2} {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.11

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} + 10 \cdot (4 {(- 3 x^{2})}^{3}) + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.12

Multiply $10$ by $4$ .

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} + 40 {(- 3 x^{2})}^{3} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.13

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} + 40 ({(- 3)}^{3} {(x^{2})}^{3}) + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.14

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} + 40 (- 27 {(x^{2})}^{3}) + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.15

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

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Step 4.1.15.1

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} + 40 (- 27 x^{2 \cdot 3}) + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.15.2

Multiply $2$ by $3$ .

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} + 40 (- 27 x^{6}) + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.16

Multiply $- 27$ by $40$ .

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} - 1080 x^{6} + 5 \cdot (- 2 {(- 3 x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.17

Multiply $5$ by $- 2$ .

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} - 1080 x^{6} - 10 {(- 3 x^{2})}^{4} + {(- 3 x^{2})}^{5}$

Step 4.1.18

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} - 1080 x^{6} - 10 ({(- 3)}^{4} {(x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.19

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} - 1080 x^{6} - 10 (81 {(x^{2})}^{4}) + {(- 3 x^{2})}^{5}$

Step 4.1.20

Multiply the exponents in ${(x^{2})}^{4}$ .

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Step 4.1.20.1

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} - 1080 x^{6} - 10 (81 x^{2 \cdot 4}) + {(- 3 x^{2})}^{5}$

Step 4.1.20.2

Multiply $2$ by $4$ .

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} - 1080 x^{6} - 10 (81 x^{8}) + {(- 3 x^{2})}^{5}$

Step 4.1.21

Multiply $81$ by $- 10$ .

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} - 1080 x^{6} - 810 x^{8} + {(- 3 x^{2})}^{5}$

Step 4.1.22

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} - 1080 x^{6} - 810 x^{8} + {(- 3)}^{5} {(x^{2})}^{5}$

Step 4.1.23

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} - 1080 x^{6} - 810 x^{8} - 243 {(x^{2})}^{5}$

Step 4.1.24

Multiply the exponents in ${(x^{2})}^{5}$ .

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Step 4.1.24.1

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

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} - 1080 x^{6} - 810 x^{8} - 243 x^{2 \cdot 5}$

Step 4.1.24.2

Multiply $2$ by $5$ .

$f (- 2 - 3 x^{2}) = - 32 - 240 x^{2} - 720 x^{4} - 1080 x^{6} - 810 x^{8} - 243 x^{10}$

Step 4.2

Simplify the expression.

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Step 4.2.1

Move $- 32$ .

$f (- 2 - 3 x^{2}) = - 240 x^{2} - 720 x^{4} - 1080 x^{6} - 810 x^{8} - 243 x^{10} - 32$

Step 4.2.2

Move $- 240 x^{2}$ .

$f (- 2 - 3 x^{2}) = - 720 x^{4} - 1080 x^{6} - 810 x^{8} - 243 x^{10} - 240 x^{2} - 32$

Step 4.2.3

Move $- 720 x^{4}$ .

$f (- 2 - 3 x^{2}) = - 1080 x^{6} - 810 x^{8} - 243 x^{10} - 720 x^{4} - 240 x^{2} - 32$

Step 4.2.4

Move $- 1080 x^{6}$ .

$f (- 2 - 3 x^{2}) = - 810 x^{8} - 243 x^{10} - 1080 x^{6} - 720 x^{4} - 240 x^{2} - 32$

Step 4.2.5

Reorder $- 810 x^{8}$ and $- 243 x^{10}$ .

$f (- 2 - 3 x^{2}) = - 243 x^{10} - 810 x^{8} - 1080 x^{6} - 720 x^{4} - 240 x^{2} - 32$