Algebra Examples

$f (x) = {(x + 4)}^{2} (x + 2) {(x - 2)}^{2}$

Step 1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 1.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$f (x) = (x + 4) (x + 4) (x + 2) {(x - 2)}^{2}$

Step 1.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = (x (x + 4) + 4 (x + 4)) (x + 2) {(x - 2)}^{2}$

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 1.3.1.2

Move $4$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $4$ by $4$ .

$f (x) = (x^{2} + 4 x + 4 x + 16) (x + 2) {(x - 2)}^{2}$

Step 1.3.2

Add $4 x$ and $4 x$ .

$f (x) = (x^{2} + 8 x + 16) (x + 2) {(x - 2)}^{2}$

Step 1.4

Expand $(x^{2} + 8 x + 16) (x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = (x^{2} x + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.5

Simplify terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

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

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Step 1.5.1.1.1

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

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Step 1.5.1.1.1.1

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

$f (x) = (x^{2} x + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.5.1.1.1.2

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

$f (x) = (x^{2 + 1} + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.5.1.1.2

Add $2$ and $1$ .

$f (x) = (x^{3} + x^{2} \cdot 2 + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.5.1.2

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

$f (x) = (x^{3} + 2 \cdot x^{2} + 8 x \cdot x + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.5.1.3

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

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Step 1.5.1.3.1

Move $x$ .

$f (x) = (x^{3} + 2 x^{2} + 8 (x \cdot x) + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.5.1.3.2

Multiply $x$ by $x$ .

$f (x) = (x^{3} + 2 x^{2} + 8 x^{2} + 8 x \cdot 2 + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.5.1.4

Multiply $2$ by $8$ .

$f (x) = (x^{3} + 2 x^{2} + 8 x^{2} + 16 x + 16 x + 16 \cdot 2) {(x - 2)}^{2}$

Step 1.5.1.5

Multiply $16$ by $2$ .

$f (x) = (x^{3} + 2 x^{2} + 8 x^{2} + 16 x + 16 x + 32) {(x - 2)}^{2}$

Step 1.5.2

Simplify by adding terms.

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Step 1.5.2.1

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

$f (x) = (x^{3} + 10 x^{2} + 16 x + 16 x + 32) {(x - 2)}^{2}$

Step 1.5.2.2

Add $16 x$ and $16 x$ .

$f (x) = (x^{3} + 10 x^{2} + 32 x + 32) {(x - 2)}^{2}$

Step 1.5.2.3

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

$f (x) = (x^{3} + 10 x^{2} + 32 x + 32) ((x - 2) (x - 2))$

Step 1.6

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

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Step 1.6.1

Apply the distributive property.

$f (x) = (x^{3} + 10 x^{2} + 32 x + 32) (x (x - 2) - 2 (x - 2))$

Step 1.6.2

Apply the distributive property.

$f (x) = (x^{3} + 10 x^{2} + 32 x + 32) (x \cdot x + x \cdot - 2 - 2 (x - 2))$

Step 1.6.3

Apply the distributive property.

$f (x) = (x^{3} + 10 x^{2} + 32 x + 32) (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $x$ by $x$ .

$f (x) = (x^{3} + 10 x^{2} + 32 x + 32) (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2)$

Step 1.7.1.2

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

$f (x) = (x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2)$

Step 1.7.1.3

Multiply $- 2$ by $- 2$ .

$f (x) = (x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 2 x - 2 x + 4)$

Step 1.7.2

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

$f (x) = (x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 4 x + 4)$

Step 1.8

Expand $(x^{3} + 10 x^{2} + 32 x + 32) (x^{2} - 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x \cdot 4 + 32 x^{2} + 32 (- 4 x) + 32 \cdot 4$

Step 1.9

Simplify terms.

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Step 1.9.1

Combine the opposite terms in $x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x \cdot 4 + 32 x^{2} + 32 (- 4 x) + 32 \cdot 4$ .

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Step 1.9.1.1

Reorder the factors in the terms $32 x \cdot 4$ and $32 (- 4 x)$ .

$f (x) = x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 4 \cdot (32 x) + 32 x^{2} - 4 \cdot (32 x) + 32 \cdot 4$

Step 1.9.1.2

Subtract $4 \cdot 32 x$ from $4 \cdot 32 x$ .

$f (x) = x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 0 + 32 \cdot 4$

Step 1.9.1.3

Add $x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2}$ and $0$ .

$f (x) = x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2

Simplify each term.

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Step 1.9.2.1

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

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Step 1.9.2.1.1

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

$f (x) = x^{3 + 2} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.1.2

Add $3$ and $2$ .

$f (x) = x^{5} + x^{3} (- 4 x) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.2

Rewrite using the commutative property of multiplication.

$f (x) = x^{5} - 4 x^{3} x + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.3

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

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Step 1.9.2.3.1

Move $x$ .

$f (x) = x^{5} - 4 (x \cdot x^{3}) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.3.2

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

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Step 1.9.2.3.2.1

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

$f (x) = x^{5} - 4 (x \cdot x^{3}) + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.3.2.2

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

$f (x) = x^{5} - 4 x^{1 + 3} + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.3.3

Add $1$ and $3$ .

$f (x) = x^{5} - 4 x^{4} + x^{3} \cdot 4 + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.4

Move $4$ to the left of $x^{3}$ .

$f (x) = x^{5} - 4 x^{4} + 4 \cdot x^{3} + 10 x^{2} x^{2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.5

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

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Step 1.9.2.5.1

Move $x^{2}$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 (x^{2} x^{2}) + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.5.2

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

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{2 + 2} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.5.3

Add $2$ and $2$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 x^{2} (- 4 x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.6

Rewrite using the commutative property of multiplication.

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot (- 4 x^{2} x) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.7

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

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Step 1.9.2.7.1

Move $x$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot (- 4 (x \cdot x^{2})) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.7.2

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

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Step 1.9.2.7.2.1

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

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot (- 4 (x \cdot x^{2})) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.7.2.2

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

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot (- 4 x^{1 + 2}) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.7.3

Add $1$ and $2$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} + 10 \cdot (- 4 x^{3}) + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.8

Multiply $10$ by $- 4$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 10 x^{2} \cdot 4 + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.9

Multiply $4$ by $10$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x \cdot x^{2} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.10

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

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Step 1.9.2.10.1

Move $x^{2}$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 (x^{2} x) + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.10.2

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

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Step 1.9.2.10.2.1

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

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 (x^{2} x) + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.10.2.2

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

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{2 + 1} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.10.3

Add $2$ and $1$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 x (- 4 x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.11

Rewrite using the commutative property of multiplication.

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 \cdot (- 4 x \cdot x) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.12

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

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Step 1.9.2.12.1

Move $x$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 \cdot (- 4 (x \cdot x)) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.12.2

Multiply $x$ by $x$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} + 32 \cdot (- 4 x^{2}) + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.13

Multiply $32$ by $- 4$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 32 \cdot 4$

Step 1.9.2.14

Multiply $32$ by $4$ .

$f (x) = x^{5} - 4 x^{4} + 4 x^{3} + 10 x^{4} - 40 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 128$

Step 1.9.3

Simplify by adding terms.

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Step 1.9.3.1

Add $- 4 x^{4}$ and $10 x^{4}$ .

$f (x) = x^{5} + 6 x^{4} + 4 x^{3} - 40 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 128$

Step 1.9.3.2

Subtract $40 x^{3}$ from $4 x^{3}$ .

$f (x) = x^{5} + 6 x^{4} - 36 x^{3} + 40 x^{2} + 32 x^{3} - 128 x^{2} + 32 x^{2} + 128$

Step 1.9.3.3

Add $- 36 x^{3}$ and $32 x^{3}$ .

$f (x) = x^{5} + 6 x^{4} - 4 x^{3} + 40 x^{2} - 128 x^{2} + 32 x^{2} + 128$

Step 1.9.3.4

Subtract $128 x^{2}$ from $40 x^{2}$ .

$f (x) = x^{5} + 6 x^{4} - 4 x^{3} - 88 x^{2} + 32 x^{2} + 128$

Step 1.9.3.5

Add $- 88 x^{2}$ and $32 x^{2}$ .

$f (x) = x^{5} + 6 x^{4} - 4 x^{3} - 56 x^{2} + 128$

Step 2

The degree of a polynomial is the highest degree of its terms.

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Step 2.1

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$x^{5} \to 5$

$6 x^{4} \to 4$

$- 4 x^{3} \to 3$

$- 56 x^{2} \to 2$

$128 \to 0$

Step 2.2

The largest exponent is the degree of the polynomial.

$5$

Step 3

The leading term in a polynomial is the term with the highest degree.

$x^{5}$

Step 4

The leading coefficient of a polynomial is the coefficient of the leading term.

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Step 4.1

The leading term in a polynomial is the term with the highest degree.

$x^{5}$

Step 4.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$1$

Step 5

List the results.

Polynomial Degree: $5$

Leading Term: $x^{5}$

Leading Coefficient: $1$