Simplify (1-2x)^3-7x(x^2-2)

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

Step 1

Simplify each term.

Use the Binomial Theorem.

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

Simplify each term.

One to any power is one.

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

One to any power is one.

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

Multiply $3$ by $1$ .

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

Multiply $- 2$ by $3$ .

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

Multiply $3$ by $1$ .

$1 - 6 x + 3 {(- 2 x)}^{2} + {(- 2 x)}^{3} - 7 x (x^{2} - 2)$

Apply the product rule to $- 2 x$ .

$1 - 6 x + 3 ({(- 2)}^{2} x^{2}) + {(- 2 x)}^{3} - 7 x (x^{2} - 2)$

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

$1 - 6 x + 3 (4 x^{2}) + {(- 2 x)}^{3} - 7 x (x^{2} - 2)$

Multiply $4$ by $3$ .

$1 - 6 x + 12 x^{2} + {(- 2 x)}^{3} - 7 x (x^{2} - 2)$

Apply the product rule to $- 2 x$ .

$1 - 6 x + 12 x^{2} + {(- 2)}^{3} x^{3} - 7 x (x^{2} - 2)$

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

$1 - 6 x + 12 x^{2} - 8 x^{3} - 7 x (x^{2} - 2)$

Apply the distributive property.

$1 - 6 x + 12 x^{2} - 8 x^{3} - 7 x \cdot x^{2} - 7 x \cdot - 2$

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

Move $x^{2}$ .

$1 - 6 x + 12 x^{2} - 8 x^{3} - 7 (x^{2} x) - 7 x \cdot - 2$

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

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

$1 - 6 x + 12 x^{2} - 8 x^{3} - 7 (x^{2} x^{1}) - 7 x \cdot - 2$

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

$1 - 6 x + 12 x^{2} - 8 x^{3} - 7 x^{2 + 1} - 7 x \cdot - 2$

Add $2$ and $1$ .

$1 - 6 x + 12 x^{2} - 8 x^{3} - 7 x^{3} - 7 x \cdot - 2$

Multiply $- 2$ by $- 7$ .

$1 - 6 x + 12 x^{2} - 8 x^{3} - 7 x^{3} + 14 x$

Step 2

Simplify by adding terms.

Add $- 6 x$ and $14 x$ .

$1 + 12 x^{2} - 8 x^{3} - 7 x^{3} + 8 x$

Subtract $7 x^{3}$ from $- 8 x^{3}$ .

$1 + 12 x^{2} - 15 x^{3} + 8 x$

Simplify the expression.

Move $1$ .

$12 x^{2} - 15 x^{3} + 8 x + 1$

Reorder $12 x^{2}$ and $- 15 x^{3}$ .

$- 15 x^{3} + 12 x^{2} + 8 x + 1$

Simplify (1-2x)^3-7x(x^2-2)

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