Algebra Examples

$f (x) = - x^{2} {(x + 8)}^{3} (x^{2} - 1)$

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Simplify terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $8$ by $3$ .

$- x^{2} (x^{3} + 24 x^{2} + 3 x \cdot 8^{2} + 8^{3}) (x^{2} - 1)$

Step 1.2.1.2

Raise $8$ to the power of $2$ .

$- x^{2} (x^{3} + 24 x^{2} + 3 x \cdot 64 + 8^{3}) (x^{2} - 1)$

Step 1.2.1.3

Multiply $64$ by $3$ .

$- x^{2} (x^{3} + 24 x^{2} + 192 x + 8^{3}) (x^{2} - 1)$

Step 1.2.1.4

Raise $8$ to the power of $3$ .

$- x^{2} (x^{3} + 24 x^{2} + 192 x + 512) (x^{2} - 1)$

Step 1.2.2

Apply the distributive property.

$(- x^{2} x^{3} - x^{2} (24 x^{2}) - x^{2} (192 x) - x^{2} \cdot 512) (x^{2} - 1)$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 1.3.1.1

Move $x^{3}$ .

$(- (x^{3} x^{2}) - x^{2} (24 x^{2}) - x^{2} (192 x) - x^{2} \cdot 512) (x^{2} - 1)$

Step 1.3.1.2

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

$(- x^{3 + 2} - x^{2} (24 x^{2}) - x^{2} (192 x) - x^{2} \cdot 512) (x^{2} - 1)$

Step 1.3.1.3

Add $3$ and $2$ .

$(- x^{5} - x^{2} (24 x^{2}) - x^{2} (192 x) - x^{2} \cdot 512) (x^{2} - 1)$

Step 1.3.2

Rewrite using the commutative property of multiplication.

$(- x^{5} - 1 \cdot 24 x^{2} x^{2} - x^{2} (192 x) - x^{2} \cdot 512) (x^{2} - 1)$

Step 1.3.3

Rewrite using the commutative property of multiplication.

$(- x^{5} - 1 \cdot 24 x^{2} x^{2} - 1 \cdot 192 x^{2} x - x^{2} \cdot 512) (x^{2} - 1)$

Step 1.3.4

Multiply $512$ by $- 1$ .

$(- x^{5} - 1 \cdot 24 x^{2} x^{2} - 1 \cdot 192 x^{2} x - 512 x^{2}) (x^{2} - 1)$

Step 1.4

Simplify each term.

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Step 1.4.1

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

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Step 1.4.1.1

Move $x^{2}$ .

$(- x^{5} - 1 \cdot 24 (x^{2} x^{2}) - 1 \cdot 192 x^{2} x - 512 x^{2}) (x^{2} - 1)$

Step 1.4.1.2

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

$(- x^{5} - 1 \cdot 24 x^{2 + 2} - 1 \cdot 192 x^{2} x - 512 x^{2}) (x^{2} - 1)$

Step 1.4.1.3

Add $2$ and $2$ .

$(- x^{5} - 1 \cdot 24 x^{4} - 1 \cdot 192 x^{2} x - 512 x^{2}) (x^{2} - 1)$

Step 1.4.2

Multiply $- 1$ by $24$ .

$(- x^{5} - 24 x^{4} - 1 \cdot 192 x^{2} x - 512 x^{2}) (x^{2} - 1)$

Step 1.4.3

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

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Step 1.4.3.1

Move $x$ .

$(- x^{5} - 24 x^{4} - 1 \cdot 192 (x \cdot x^{2}) - 512 x^{2}) (x^{2} - 1)$

Step 1.4.3.2

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

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Step 1.4.3.2.1

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

$(- x^{5} - 24 x^{4} - 1 \cdot 192 (x^{1} x^{2}) - 512 x^{2}) (x^{2} - 1)$

Step 1.4.3.2.2

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

$(- x^{5} - 24 x^{4} - 1 \cdot 192 x^{1 + 2} - 512 x^{2}) (x^{2} - 1)$

Step 1.4.3.3

Add $1$ and $2$ .

$(- x^{5} - 24 x^{4} - 1 \cdot 192 x^{3} - 512 x^{2}) (x^{2} - 1)$

Step 1.4.4

Multiply $- 1$ by $192$ .

$(- x^{5} - 24 x^{4} - 192 x^{3} - 512 x^{2}) (x^{2} - 1)$

Step 1.5

Expand $(- x^{5} - 24 x^{4} - 192 x^{3} - 512 x^{2}) (x^{2} - 1)$ by multiplying each term in the first expression by each term in the second expression.

$- x^{5} x^{2} - x^{5} \cdot - 1 - 24 x^{4} x^{2} - 24 x^{4} \cdot - 1 - 192 x^{3} x^{2} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6

Simplify terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

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Step 1.6.1.1.1

Move $x^{2}$ .

$- (x^{2} x^{5}) - x^{5} \cdot - 1 - 24 x^{4} x^{2} - 24 x^{4} \cdot - 1 - 192 x^{3} x^{2} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.1.2

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

$- x^{2 + 5} - x^{5} \cdot - 1 - 24 x^{4} x^{2} - 24 x^{4} \cdot - 1 - 192 x^{3} x^{2} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.1.3

Add $2$ and $5$ .

$- x^{7} - x^{5} \cdot - 1 - 24 x^{4} x^{2} - 24 x^{4} \cdot - 1 - 192 x^{3} x^{2} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.2

Multiply $- x^{5} \cdot - 1$ .

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Step 1.6.1.2.1

Multiply $- 1$ by $- 1$ .

$- x^{7} + 1 x^{5} - 24 x^{4} x^{2} - 24 x^{4} \cdot - 1 - 192 x^{3} x^{2} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.2.2

Multiply $x^{5}$ by $1$ .

$- x^{7} + x^{5} - 24 x^{4} x^{2} - 24 x^{4} \cdot - 1 - 192 x^{3} x^{2} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.3

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 1.6.1.3.1

Move $x^{2}$ .

$- x^{7} + x^{5} - 24 (x^{2} x^{4}) - 24 x^{4} \cdot - 1 - 192 x^{3} x^{2} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.3.2

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

$- x^{7} + x^{5} - 24 x^{2 + 4} - 24 x^{4} \cdot - 1 - 192 x^{3} x^{2} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.3.3

Add $2$ and $4$ .

$- x^{7} + x^{5} - 24 x^{6} - 24 x^{4} \cdot - 1 - 192 x^{3} x^{2} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.4

Multiply $- 1$ by $- 24$ .

$- x^{7} + x^{5} - 24 x^{6} + 24 x^{4} - 192 x^{3} x^{2} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.5

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 1.6.1.5.1

Move $x^{2}$ .

$- x^{7} + x^{5} - 24 x^{6} + 24 x^{4} - 192 (x^{2} x^{3}) - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.5.2

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

$- x^{7} + x^{5} - 24 x^{6} + 24 x^{4} - 192 x^{2 + 3} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.5.3

Add $2$ and $3$ .

$- x^{7} + x^{5} - 24 x^{6} + 24 x^{4} - 192 x^{5} - 192 x^{3} \cdot - 1 - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.6

Multiply $- 1$ by $- 192$ .

$- x^{7} + x^{5} - 24 x^{6} + 24 x^{4} - 192 x^{5} + 192 x^{3} - 512 x^{2} x^{2} - 512 x^{2} \cdot - 1$

Step 1.6.1.7

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

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Step 1.6.1.7.1

Move $x^{2}$ .

$- x^{7} + x^{5} - 24 x^{6} + 24 x^{4} - 192 x^{5} + 192 x^{3} - 512 (x^{2} x^{2}) - 512 x^{2} \cdot - 1$

Step 1.6.1.7.2

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

$- x^{7} + x^{5} - 24 x^{6} + 24 x^{4} - 192 x^{5} + 192 x^{3} - 512 x^{2 + 2} - 512 x^{2} \cdot - 1$

Step 1.6.1.7.3

Add $2$ and $2$ .

$- x^{7} + x^{5} - 24 x^{6} + 24 x^{4} - 192 x^{5} + 192 x^{3} - 512 x^{4} - 512 x^{2} \cdot - 1$

Step 1.6.1.8

Multiply $- 1$ by $- 512$ .

$- x^{7} + x^{5} - 24 x^{6} + 24 x^{4} - 192 x^{5} + 192 x^{3} - 512 x^{4} + 512 x^{2}$

Step 1.6.2

Simplify by adding terms.

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Step 1.6.2.1

Subtract $192 x^{5}$ from $x^{5}$ .

$- x^{7} - 191 x^{5} - 24 x^{6} + 24 x^{4} + 192 x^{3} - 512 x^{4} + 512 x^{2}$

Step 1.6.2.2

Subtract $512 x^{4}$ from $24 x^{4}$ .

$- x^{7} - 191 x^{5} - 24 x^{6} - 488 x^{4} + 192 x^{3} + 512 x^{2}$

Step 2

The largest exponent is the degree of the polynomial.

$7$