Álgebra Ejemplos

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

Paso 1

Establece la función de resultado compuesta.

$f (g (x))$

Paso 2

Evalúa $f (5 x^{2} + x - 1)$ mediante la sustitución del valor de $g$ en $f$ .

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

Paso 3

Simplifica cada término.

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Paso 3.1

Usa el teorema multinomial.

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

Paso 3.2

Simplifica cada término.

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Paso 3.2.1

Aplica la regla del producto a $5 x^{2}$ .

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

Paso 3.2.2

Eleva $5$ a la potencia de $3$ .

$f (5 x^{2} + x - 1) = 3 (125 {(x^{2})}^{3} + 3 {(5 x^{2})}^{2} x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.3

Multiplica los exponentes en ${(x^{2})}^{3}$ .

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Paso 3.2.3.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{2 \cdot 3} + 3 {(5 x^{2})}^{2} x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.3.2

Multiplica $2$ por $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 {(5 x^{2})}^{2} x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.4

Aplica la regla del producto a $5 x^{2}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (5^{2} {(x^{2})}^{2}) x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.5

Eleva $5$ a la potencia de $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 {(x^{2})}^{2}) x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.6

Multiplica los exponentes en ${(x^{2})}^{2}$ .

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Paso 3.2.6.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 x^{2 \cdot 2}) x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.6.2

Multiplica $2$ por $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 x^{4}) x + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.7

Multiplica $x^{4}$ por $x$ sumando los exponentes.

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Paso 3.2.7.1

Mueve $x$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 (x \cdot x^{4})) + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.7.2

Multiplica $x$ por $x^{4}$ .

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Paso 3.2.7.2.1

Eleva $x$ a la potencia de $1$ .

Paso 3.2.7.2.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 x^{1 + 4}) + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.7.3

Suma $1$ y $4$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 3 (25 x^{5}) + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.8

Multiplica $25$ por $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 3 (5 x^{2}) x^{2} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.9

Multiplica $x^{2}$ por $x^{2}$ sumando los exponentes.

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Paso 3.2.9.1

Mueve $x^{2}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 3 (5 (x^{2} x^{2})) + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.9.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 3 (5 x^{2 + 2}) + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.9.3

Suma $2$ y $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 3 (5 x^{4}) + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.10

Multiplica $5$ por $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 3 {(5 x^{2})}^{2} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.11

Aplica la regla del producto a $5 x^{2}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 3 (5^{2} {(x^{2})}^{2}) \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.12

Eleva $5$ a la potencia de $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 3 (25 {(x^{2})}^{2}) \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.13

Multiplica los exponentes en ${(x^{2})}^{2}$ .

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Paso 3.2.13.1

Aplica la regla de la potencia y multiplica los exponentes, ${(a^{m})}^{n} = a^{m n}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 3 (25 x^{2 \cdot 2}) \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.13.2

Multiplica $2$ por $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 3 (25 x^{4}) \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.14

Multiplica $25$ por $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} + 75 x^{4} \cdot - 1 + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.15

Multiplica $- 1$ por $75$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} + 6 (5 x^{2}) x \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.16

Multiplica $x^{2}$ por $x$ sumando los exponentes.

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Paso 3.2.16.1

Mueve $x$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} + 6 (5 (x \cdot x^{2})) \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.16.2

Multiplica $x$ por $x^{2}$ .

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Paso 3.2.16.2.1

Eleva $x$ a la potencia de $1$ .

Paso 3.2.16.2.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} + 6 (5 x^{1 + 2}) \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.16.3

Suma $1$ y $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} + 6 (5 x^{3}) \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.17

Multiplica $5$ por $6$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} + 30 x^{3} \cdot - 1 + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.18

Multiplica $- 1$ por $30$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} + 3 x^{2} \cdot - 1 + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.19

Multiplica $- 1$ por $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 3 (5 x^{2}) {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.20

Multiplica $5$ por $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} {(- 1)}^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.21

Eleva $- 1$ a la potencia de $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} \cdot 1 + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.22

Multiplica $15$ por $1$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.23

Eleva $- 1$ a la potencia de $2$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} + 3 x \cdot 1 + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.24

Multiplica $3$ por $1$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} + 3 x + {(- 1)}^{3}) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.2.25

Eleva $- 1$ a la potencia de $3$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} + 15 x^{4} + x^{3} - 75 x^{4} - 30 x^{3} - 3 x^{2} + 15 x^{2} + 3 x - 1) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.3

Resta $75 x^{4}$ de $15 x^{4}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} - 60 x^{4} + x^{3} - 30 x^{3} - 3 x^{2} + 15 x^{2} + 3 x - 1) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.4

Resta $30 x^{3}$ de $x^{3}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} - 60 x^{4} - 29 x^{3} - 3 x^{2} + 15 x^{2} + 3 x - 1) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.5

Suma $- 3 x^{2}$ y $15 x^{2}$ .

$f (5 x^{2} + x - 1) = 3 (125 x^{6} + 75 x^{5} - 60 x^{4} - 29 x^{3} + 12 x^{2} + 3 x - 1) - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.6

Aplica la propiedad distributiva.

$f (5 x^{2} + x - 1) = 3 (125 x^{6}) + 3 (75 x^{5}) + 3 (- 60 x^{4}) + 3 (- 29 x^{3}) + 3 (12 x^{2}) + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.7

Simplifica.

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Paso 3.7.1

Multiplica $125$ por $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 3 (75 x^{5}) + 3 (- 60 x^{4}) + 3 (- 29 x^{3}) + 3 (12 x^{2}) + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.7.2

Multiplica $75$ por $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} + 3 (- 60 x^{4}) + 3 (- 29 x^{3}) + 3 (12 x^{2}) + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.7.3

Multiplica $- 60$ por $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} + 3 (- 29 x^{3}) + 3 (12 x^{2}) + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.7.4

Multiplica $- 29$ por $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 3 (12 x^{2}) + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.7.5

Multiplica $12$ por $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 3 (3 x) + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.7.6

Multiplica $3$ por $3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x + 3 \cdot - 1 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.7.7

Multiplica $3$ por $- 1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 {(5 x^{2} + x - 1)}^{2} + 7 (5 x^{2} + x - 1) - 5$

Paso 3.8

Reescribe ${(5 x^{2} + x - 1)}^{2}$ como $(5 x^{2} + x - 1) (5 x^{2} + x - 1)$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 ((5 x^{2} + x - 1) (5 x^{2} + x - 1)) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.9

Expande $(5 x^{2} + x - 1) (5 x^{2} + x - 1)$ mediante la multiplicación de cada término de la primera expresión por cada término de la segunda expresión.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (5 x^{2} (5 x^{2}) + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10

Simplifica cada término.

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Paso 3.10.1

Reescribe con la propiedad conmutativa de la multiplicación.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (5 \cdot (5 x^{2} x^{2}) + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.2

Multiplica $x^{2}$ por $x^{2}$ sumando los exponentes.

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Paso 3.10.2.1

Mueve $x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (5 \cdot (5 (x^{2} x^{2})) + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.2.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (5 \cdot (5 x^{2 + 2}) + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.2.3

Suma $2$ y $2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (5 \cdot (5 x^{4}) + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.3

Multiplica $5$ por $5$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{2} x + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.4

Multiplica $x^{2}$ por $x$ sumando los exponentes.

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Paso 3.10.4.1

Mueve $x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 (x \cdot x^{2}) + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.4.2

Multiplica $x$ por $x^{2}$ .

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Paso 3.10.4.2.1

Eleva $x$ a la potencia de $1$ .

Paso 3.10.4.2.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{1 + 2} + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.4.3

Suma $1$ y $2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} + 5 x^{2} \cdot - 1 + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.5

Multiplica $- 1$ por $5$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + x (5 x^{2}) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.6

Reescribe con la propiedad conmutativa de la multiplicación.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x \cdot x^{2} + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.7

Multiplica $x$ por $x^{2}$ sumando los exponentes.

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Paso 3.10.7.1

Mueve $x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 (x^{2} x) + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.7.2

Multiplica $x^{2}$ por $x$ .

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Paso 3.10.7.2.1

Eleva $x$ a la potencia de $1$ .

Paso 3.10.7.2.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{2 + 1} + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.7.3

Suma $2$ y $1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x \cdot x + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.8

Multiplica $x$ por $x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} + x \cdot - 1 - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.9

Mueve $- 1$ a la izquierda de $x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} - 1 \cdot x - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.10

Reescribe $- 1 x$ como $- x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} - x - 1 (5 x^{2}) - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.11

Multiplica $5$ por $- 1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} - x - 5 x^{2} - 1 x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.12

Reescribe $- 1 x$ como $- x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} - x - 5 x^{2} - x - 1 \cdot - 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.10.13

Multiplica $- 1$ por $- 1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 5 x^{3} - 5 x^{2} + 5 x^{3} + x^{2} - x - 5 x^{2} - x + 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.11

Suma $5 x^{3}$ y $5 x^{3}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 10 x^{3} - 5 x^{2} + x^{2} - x - 5 x^{2} - x + 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.12

Suma $- 5 x^{2}$ y $x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 10 x^{3} - 4 x^{2} - x - 5 x^{2} - x + 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.13

Resta $5 x^{2}$ de $- 4 x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 10 x^{3} - 9 x^{2} - x - x + 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.14

Resta $x$ de $- x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4} + 10 x^{3} - 9 x^{2} - 2 x + 1) + 7 (5 x^{2} + x - 1) - 5$

Paso 3.15

Aplica la propiedad distributiva.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 2 (25 x^{4}) - 2 (10 x^{3}) - 2 (- 9 x^{2}) - 2 (- 2 x) - 2 \cdot 1 + 7 (5 x^{2} + x - 1) - 5$

Paso 3.16

Simplifica.

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Paso 3.16.1

Multiplica $25$ por $- 2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 2 (10 x^{3}) - 2 (- 9 x^{2}) - 2 (- 2 x) - 2 \cdot 1 + 7 (5 x^{2} + x - 1) - 5$

Paso 3.16.2

Multiplica $10$ por $- 2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} - 2 (- 9 x^{2}) - 2 (- 2 x) - 2 \cdot 1 + 7 (5 x^{2} + x - 1) - 5$

Paso 3.16.3

Multiplica $- 9$ por $- 2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} - 2 (- 2 x) - 2 \cdot 1 + 7 (5 x^{2} + x - 1) - 5$

Paso 3.16.4

Multiplica $- 2$ por $- 2$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} + 4 x - 2 \cdot 1 + 7 (5 x^{2} + x - 1) - 5$

Paso 3.16.5

Multiplica $- 2$ por $1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} + 4 x - 2 + 7 (5 x^{2} + x - 1) - 5$

Paso 3.17

Aplica la propiedad distributiva.

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} + 4 x - 2 + 7 (5 x^{2}) + 7 x + 7 \cdot - 1 - 5$

Paso 3.18

Simplifica.

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Paso 3.18.1

Multiplica $5$ por $7$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} + 4 x - 2 + 35 x^{2} + 7 x + 7 \cdot - 1 - 5$

Paso 3.18.2

Multiplica $7$ por $- 1$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 180 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 50 x^{4} - 20 x^{3} + 18 x^{2} + 4 x - 2 + 35 x^{2} + 7 x - 7 - 5$

Paso 4

Simplifica mediante la adición de términos.

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Paso 4.1

Resta $50 x^{4}$ de $- 180 x^{4}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 87 x^{3} + 36 x^{2} + 9 x - 3 - 20 x^{3} + 18 x^{2} + 4 x - 2 + 35 x^{2} + 7 x - 7 - 5$

Paso 4.2

Resta $20 x^{3}$ de $- 87 x^{3}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 36 x^{2} + 9 x - 3 + 18 x^{2} + 4 x - 2 + 35 x^{2} + 7 x - 7 - 5$

Paso 4.3

Suma $36 x^{2}$ y $18 x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 54 x^{2} + 9 x - 3 + 4 x - 2 + 35 x^{2} + 7 x - 7 - 5$

Paso 4.4

Suma $54 x^{2}$ y $35 x^{2}$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 9 x - 3 + 4 x - 2 + 7 x - 7 - 5$

Paso 4.5

Suma $9 x$ y $4 x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 13 x - 3 - 2 + 7 x - 7 - 5$

Paso 4.6

Suma $13 x$ y $7 x$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 20 x - 3 - 2 - 7 - 5$

Paso 4.7

Simplifica mediante la resta de números.

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Paso 4.7.1

Resta $2$ de $- 3$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 20 x - 5 - 7 - 5$

Paso 4.7.2

Resta $7$ de $- 5$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 20 x - 12 - 5$

Paso 4.7.3

Resta $5$ de $- 12$ .

$f (5 x^{2} + x - 1) = 375 x^{6} + 225 x^{5} - 230 x^{4} - 107 x^{3} + 89 x^{2} + 20 x - 17$