Algebra Examples

$- 2 x (x^{3} - 6 x^{2} + 6) + 4 x^{3} - (5 x^{4} + 10 x)$

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$- 2 x \cdot x^{3} - 2 x (- 6 x^{2}) - 2 x \cdot 6 + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 2.2.1.1

Move $x^{3}$ .

$- 2 (x^{3} x) - 2 x (- 6 x^{2}) - 2 x \cdot 6 + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.2.1.2

Multiply $x^{3}$ by $x$ .

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Step 2.2.1.2.1

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

$- 2 (x^{3} x^{1}) - 2 x (- 6 x^{2}) - 2 x \cdot 6 + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.2.1.2.2

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

$- 2 x^{3 + 1} - 2 x (- 6 x^{2}) - 2 x \cdot 6 + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.2.1.3

Add $3$ and $1$ .

$- 2 x^{4} - 2 x (- 6 x^{2}) - 2 x \cdot 6 + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.2.2

Rewrite using the commutative property of multiplication.

$- 2 x^{4} - 2 \cdot - 6 x \cdot x^{2} - 2 x \cdot 6 + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.2.3

Multiply $6$ by $- 2$ .

$- 2 x^{4} - 2 \cdot - 6 x \cdot x^{2} - 12 x + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.3

Simplify each term.

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Step 2.3.1

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

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Step 2.3.1.1

Move $x^{2}$ .

$- 2 x^{4} - 2 \cdot - 6 (x^{2} x) - 12 x + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

$- 2 x^{4} - 2 \cdot - 6 (x^{2} x^{1}) - 12 x + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.3.1.2.2

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

$- 2 x^{4} - 2 \cdot - 6 x^{2 + 1} - 12 x + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.3.1.3

Add $2$ and $1$ .

$- 2 x^{4} - 2 \cdot - 6 x^{3} - 12 x + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.3.2

Multiply $- 2$ by $- 6$ .

$- 2 x^{4} + 12 x^{3} - 12 x + 4 x^{3} - (5 x^{4} + 10 x)$

Step 2.4

Apply the distributive property.

$- 2 x^{4} + 12 x^{3} - 12 x + 4 x^{3} - (5 x^{4}) - (10 x)$

Step 2.5

Multiply $5$ by $- 1$ .

$- 2 x^{4} + 12 x^{3} - 12 x + 4 x^{3} - 5 x^{4} - (10 x)$

Step 2.6

Multiply $10$ by $- 1$ .

$- 2 x^{4} + 12 x^{3} - 12 x + 4 x^{3} - 5 x^{4} - 10 x$

Step 3

Simplify by adding terms.

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Step 3.1

Subtract $5 x^{4}$ from $- 2 x^{4}$ .

$- 7 x^{4} + 12 x^{3} - 12 x + 4 x^{3} - 10 x$

Step 3.2

Add $12 x^{3}$ and $4 x^{3}$ .

$- 7 x^{4} + 16 x^{3} - 12 x - 10 x$

Step 3.3

Subtract $10 x$ from $- 12 x$ .

$- 7 x^{4} + 16 x^{3} - 22 x$