Algebra Examples

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

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Apply the distributive property.

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

Step 1.2

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

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Step 1.2.1

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.2

Add $3$ and $1$ .

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

Step 1.3

Move $- 2$ to the left of $x^{3}$ .

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

Step 1.4

Use the Binomial Theorem.

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

Step 1.5

Simplify each term.

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Step 1.5.1

Multiply $3$ by $1$ .

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

Step 1.5.2

One to any power is one.

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

Step 1.5.3

Multiply $3$ by $1$ .

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

Step 1.5.4

One to any power is one.

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

Step 1.6

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

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

Step 1.7

Simplify terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 1.7.1.1.1

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

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

Step 1.7.1.1.2

Add $4$ and $3$ .

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

Step 1.7.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.3

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

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Step 1.7.1.3.1

Move $x^{2}$ .

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

Step 1.7.1.3.2

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

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

Step 1.7.1.3.3

Add $2$ and $4$ .

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

Step 1.7.1.4

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.5

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

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Step 1.7.1.5.1

Move $x$ .

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

Step 1.7.1.5.2

Multiply $x$ by $x^{4}$ .

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Step 1.7.1.5.2.1

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

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

Step 1.7.1.5.2.2

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

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

Step 1.7.1.5.3

Add $1$ and $4$ .

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

Step 1.7.1.6

Multiply $x^{4}$ by $1$ .

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

Step 1.7.1.7

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

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Step 1.7.1.7.1

Move $x^{3}$ .

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

Step 1.7.1.7.2

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

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

Step 1.7.1.7.3

Add $3$ and $3$ .

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

Step 1.7.1.8

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.9

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

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Step 1.7.1.9.1

Move $x^{2}$ .

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

Step 1.7.1.9.2

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

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

Step 1.7.1.9.3

Add $2$ and $3$ .

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

Step 1.7.1.10

Multiply $- 2$ by $3$ .

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

Step 1.7.1.11

Rewrite using the commutative property of multiplication.

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

Step 1.7.1.12

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

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Step 1.7.1.12.1

Move $x$ .

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

Step 1.7.1.12.2

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

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Step 1.7.1.12.2.1

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

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

Step 1.7.1.12.2.2

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

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

Step 1.7.1.12.3

Add $1$ and $3$ .

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

Step 1.7.1.13

Multiply $- 2$ by $3$ .

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

Step 1.7.1.14

Multiply $- 2$ by $1$ .

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

Step 1.7.2

Simplify by adding terms.

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Step 1.7.2.1

Subtract $2 x^{6}$ from $3 x^{6}$ .

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

Step 1.7.2.2

Subtract $6 x^{5}$ from $3 x^{5}$ .

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

Step 1.7.2.3

Subtract $6 x^{4}$ from $x^{4}$ .

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

Step 2

The largest exponent is the degree of the polynomial.

$7$