Algebra Examples

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

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

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

Step 1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Move $- 2$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 1.3.2

Subtract $2 x$ from $- 2 x$ .

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Simplify.

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Step 1.5.1

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

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Step 1.5.1.1

Move $x^{2}$ .

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

Step 1.5.1.2

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

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Step 1.5.1.2.1

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

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

Step 1.5.1.2.2

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

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

Step 1.5.1.3

Add $2$ and $1$ .

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

Step 1.5.2

Rewrite using the commutative property of multiplication.

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

Step 1.5.3

Multiply $4$ by $2$ .

$(2 x^{3} + 2 \cdot - 4 x \cdot x + 8 x) {(x + 1)}^{2}$

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 $x$ by $x$ by adding the exponents.

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Step 1.6.1.1.1

Move $x$ .

$(2 x^{3} + 2 \cdot - 4 (x \cdot x) + 8 x) {(x + 1)}^{2}$

Step 1.6.1.1.2

Multiply $x$ by $x$ .

$(2 x^{3} + 2 \cdot - 4 x^{2} + 8 x) {(x + 1)}^{2}$

Step 1.6.1.2

Multiply $2$ by $- 4$ .

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

Step 1.6.2

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

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

Step 1.7

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.7.1

Apply the distributive property.

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

Step 1.7.2

Apply the distributive property.

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

Step 1.7.3

Apply the distributive property.

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

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Multiply $x$ by $x$ .

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

Step 1.8.1.2

Multiply $x$ by $1$ .

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

Step 1.8.1.3

Multiply $x$ by $1$ .

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

Step 1.8.1.4

Multiply $1$ by $1$ .

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

Step 1.8.2

Add $x$ and $x$ .

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

Step 1.9

Expand $(2 x^{3} - 8 x^{2} + 8 x) (x^{2} + 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.10

Simplify terms.

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Step 1.10.1

Simplify each term.

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Step 1.10.1.1

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

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Step 1.10.1.1.1

Move $x^{2}$ .

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

Step 1.10.1.1.2

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

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

Step 1.10.1.1.3

Add $2$ and $3$ .

$2 x^{5} + 2 x^{3} (2 x) + 2 x^{3} \cdot 1 - 8 x^{2} x^{2} - 8 x^{2} (2 x) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.2

Rewrite using the commutative property of multiplication.

$2 x^{5} + 2 \cdot 2 x^{3} x + 2 x^{3} \cdot 1 - 8 x^{2} x^{2} - 8 x^{2} (2 x) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.3

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

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Step 1.10.1.3.1

Move $x$ .

$2 x^{5} + 2 \cdot 2 (x \cdot x^{3}) + 2 x^{3} \cdot 1 - 8 x^{2} x^{2} - 8 x^{2} (2 x) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.3.2

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

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Step 1.10.1.3.2.1

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

$2 x^{5} + 2 \cdot 2 (x^{1} x^{3}) + 2 x^{3} \cdot 1 - 8 x^{2} x^{2} - 8 x^{2} (2 x) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.3.2.2

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

$2 x^{5} + 2 \cdot 2 x^{1 + 3} + 2 x^{3} \cdot 1 - 8 x^{2} x^{2} - 8 x^{2} (2 x) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.3.3

Add $1$ and $3$ .

$2 x^{5} + 2 \cdot 2 x^{4} + 2 x^{3} \cdot 1 - 8 x^{2} x^{2} - 8 x^{2} (2 x) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.4

Multiply $2$ by $2$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} \cdot 1 - 8 x^{2} x^{2} - 8 x^{2} (2 x) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.5

Multiply $2$ by $1$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{2} x^{2} - 8 x^{2} (2 x) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.6

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

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Step 1.10.1.6.1

Move $x^{2}$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 (x^{2} x^{2}) - 8 x^{2} (2 x) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.6.2

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

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{2 + 2} - 8 x^{2} (2 x) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.6.3

Add $2$ and $2$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 8 x^{2} (2 x) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.7

Rewrite using the commutative property of multiplication.

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 8 \cdot 2 x^{2} x - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.8

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

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Step 1.10.1.8.1

Move $x$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 8 \cdot 2 (x \cdot x^{2}) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.8.2

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

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Step 1.10.1.8.2.1

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

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 8 \cdot 2 (x^{1} x^{2}) - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.8.2.2

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

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 8 \cdot 2 x^{1 + 2} - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.8.3

Add $1$ and $2$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 8 \cdot 2 x^{3} - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.9

Multiply $- 8$ by $2$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 16 x^{3} - 8 x^{2} \cdot 1 + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.10

Multiply $- 8$ by $1$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 16 x^{3} - 8 x^{2} + 8 x \cdot x^{2} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.11

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

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Step 1.10.1.11.1

Move $x^{2}$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 16 x^{3} - 8 x^{2} + 8 (x^{2} x) + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.11.2

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

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Step 1.10.1.11.2.1

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

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 16 x^{3} - 8 x^{2} + 8 (x^{2} x^{1}) + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.11.2.2

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

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 16 x^{3} - 8 x^{2} + 8 x^{2 + 1} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.11.3

Add $2$ and $1$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 16 x^{3} - 8 x^{2} + 8 x^{3} + 8 x (2 x) + 8 x \cdot 1$

Step 1.10.1.12

Rewrite using the commutative property of multiplication.

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 16 x^{3} - 8 x^{2} + 8 x^{3} + 8 \cdot 2 x \cdot x + 8 x \cdot 1$

Step 1.10.1.13

Multiply $x$ by $x$ by adding the exponents.

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Step 1.10.1.13.1

Move $x$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 16 x^{3} - 8 x^{2} + 8 x^{3} + 8 \cdot 2 (x \cdot x) + 8 x \cdot 1$

Step 1.10.1.13.2

Multiply $x$ by $x$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 16 x^{3} - 8 x^{2} + 8 x^{3} + 8 \cdot 2 x^{2} + 8 x \cdot 1$

Step 1.10.1.14

Multiply $8$ by $2$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 16 x^{3} - 8 x^{2} + 8 x^{3} + 16 x^{2} + 8 x \cdot 1$

Step 1.10.1.15

Multiply $8$ by $1$ .

$2 x^{5} + 4 x^{4} + 2 x^{3} - 8 x^{4} - 16 x^{3} - 8 x^{2} + 8 x^{3} + 16 x^{2} + 8 x$

Step 1.10.2

Simplify by adding terms.

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Step 1.10.2.1

Subtract $8 x^{4}$ from $4 x^{4}$ .

$2 x^{5} - 4 x^{4} + 2 x^{3} - 16 x^{3} - 8 x^{2} + 8 x^{3} + 16 x^{2} + 8 x$

Step 1.10.2.2

Subtract $16 x^{3}$ from $2 x^{3}$ .

$2 x^{5} - 4 x^{4} - 14 x^{3} - 8 x^{2} + 8 x^{3} + 16 x^{2} + 8 x$

Step 1.10.2.3

Add $- 14 x^{3}$ and $8 x^{3}$ .

$2 x^{5} - 4 x^{4} - 6 x^{3} - 8 x^{2} + 16 x^{2} + 8 x$

Step 1.10.2.4

Add $- 8 x^{2}$ and $16 x^{2}$ .

$2 x^{5} - 4 x^{4} - 6 x^{3} + 8 x^{2} + 8 x$

Step 2

The largest exponent is the degree of the polynomial.

$5$