Algebra Examples

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

Step 1

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

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Step 1.1

Use the Binomial Theorem.

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

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

Apply the product rule to $2 x$ .

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

Step 1.2.1.2

Raise $2$ to the power of $3$ .

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

Step 1.2.1.3

Apply the product rule to $2 x$ .

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

Step 1.2.1.4

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

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

Step 1.2.1.5

Multiply $4$ by $3$ .

$f (x) = - 2 x^{2} (8 x^{3} + 12 x^{2} \cdot - 1 + 3 (2 x) {(- 1)}^{2} + {(- 1)}^{3}) (4 x + 3)$

Step 1.2.1.6

Multiply $- 1$ by $12$ .

$f (x) = - 2 x^{2} (8 x^{3} - 12 x^{2} + 3 (2 x) {(- 1)}^{2} + {(- 1)}^{3}) (4 x + 3)$

Step 1.2.1.7

Multiply $2$ by $3$ .

$f (x) = - 2 x^{2} (8 x^{3} - 12 x^{2} + 6 x {(- 1)}^{2} + {(- 1)}^{3}) (4 x + 3)$

Step 1.2.1.8

Raise $- 1$ to the power of $2$ .

$f (x) = - 2 x^{2} (8 x^{3} - 12 x^{2} + 6 x \cdot 1 + {(- 1)}^{3}) (4 x + 3)$

Step 1.2.1.9

Multiply $6$ by $1$ .

$f (x) = - 2 x^{2} (8 x^{3} - 12 x^{2} + 6 x + {(- 1)}^{3}) (4 x + 3)$

Step 1.2.1.10

Raise $- 1$ to the power of $3$ .

$f (x) = - 2 x^{2} (8 x^{3} - 12 x^{2} + 6 x - 1) (4 x + 3)$

Step 1.2.2

Apply the distributive property.

$f (x) = (- 2 x^{2} (8 x^{3}) - 2 x^{2} (- 12 x^{2}) - 2 x^{2} (6 x) - 2 x^{2} \cdot - 1) (4 x + 3)$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$f (x) = (- 2 \cdot (8 x^{2} x^{3}) - 2 x^{2} (- 12 x^{2}) - 2 x^{2} (6 x) - 2 x^{2} \cdot - 1) (4 x + 3)$

Step 1.3.2

Rewrite using the commutative property of multiplication.

$f (x) = (- 2 \cdot (8 x^{2} x^{3}) - 2 \cdot (- 12 x^{2} x^{2}) - 2 x^{2} (6 x) - 2 x^{2} \cdot - 1) (4 x + 3)$

Step 1.3.3

Rewrite using the commutative property of multiplication.

$f (x) = (- 2 \cdot (8 x^{2} x^{3}) - 2 \cdot (- 12 x^{2} x^{2}) - 2 \cdot (6 x^{2} x) - 2 x^{2} \cdot - 1) (4 x + 3)$

Step 1.3.4

Multiply $- 1$ by $- 2$ .

$f (x) = (- 2 \cdot (8 x^{2} x^{3}) - 2 \cdot (- 12 x^{2} x^{2}) - 2 \cdot (6 x^{2} x) + 2 x^{2}) (4 x + 3)$

Step 1.4

Simplify each term.

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Step 1.4.1

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

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Step 1.4.1.1

Move $x^{3}$ .

$f (x) = (- 2 \cdot (8 (x^{3} x^{2})) - 2 \cdot (- 12 x^{2} x^{2}) - 2 \cdot (6 x^{2} x) + 2 x^{2}) (4 x + 3)$

Step 1.4.1.2

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

$f (x) = (- 2 \cdot (8 x^{3 + 2}) - 2 \cdot (- 12 x^{2} x^{2}) - 2 \cdot (6 x^{2} x) + 2 x^{2}) (4 x + 3)$

Step 1.4.1.3

Add $3$ and $2$ .

$f (x) = (- 2 \cdot (8 x^{5}) - 2 \cdot (- 12 x^{2} x^{2}) - 2 \cdot (6 x^{2} x) + 2 x^{2}) (4 x + 3)$

Step 1.4.2

Multiply $- 2$ by $8$ .

$f (x) = (- 16 x^{5} - 2 \cdot (- 12 x^{2} x^{2}) - 2 \cdot (6 x^{2} x) + 2 x^{2}) (4 x + 3)$

Step 1.4.3

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

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Step 1.4.3.1

Move $x^{2}$ .

$f (x) = (- 16 x^{5} - 2 \cdot (- 12 (x^{2} x^{2})) - 2 \cdot (6 x^{2} x) + 2 x^{2}) (4 x + 3)$

Step 1.4.3.2

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

$f (x) = (- 16 x^{5} - 2 \cdot (- 12 x^{2 + 2}) - 2 \cdot (6 x^{2} x) + 2 x^{2}) (4 x + 3)$

Step 1.4.3.3

Add $2$ and $2$ .

$f (x) = (- 16 x^{5} - 2 \cdot (- 12 x^{4}) - 2 \cdot (6 x^{2} x) + 2 x^{2}) (4 x + 3)$

Step 1.4.4

Multiply $- 2$ by $- 12$ .

$f (x) = (- 16 x^{5} + 24 x^{4} - 2 \cdot (6 x^{2} x) + 2 x^{2}) (4 x + 3)$

Step 1.4.5

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

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Step 1.4.5.1

Move $x$ .

$f (x) = (- 16 x^{5} + 24 x^{4} - 2 \cdot (6 (x \cdot x^{2})) + 2 x^{2}) (4 x + 3)$

Step 1.4.5.2

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

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Step 1.4.5.2.1

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

$f (x) = (- 16 x^{5} + 24 x^{4} - 2 \cdot (6 (x \cdot x^{2})) + 2 x^{2}) (4 x + 3)$

Step 1.4.5.2.2

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

$f (x) = (- 16 x^{5} + 24 x^{4} - 2 \cdot (6 x^{1 + 2}) + 2 x^{2}) (4 x + 3)$

Step 1.4.5.3

Add $1$ and $2$ .

$f (x) = (- 16 x^{5} + 24 x^{4} - 2 \cdot (6 x^{3}) + 2 x^{2}) (4 x + 3)$

Step 1.4.6

Multiply $- 2$ by $6$ .

$f (x) = (- 16 x^{5} + 24 x^{4} - 12 x^{3} + 2 x^{2}) (4 x + 3)$

Step 1.5

Expand $(- 16 x^{5} + 24 x^{4} - 12 x^{3} + 2 x^{2}) (4 x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = - 16 x^{5} (4 x) - 16 x^{5} \cdot 3 + 24 x^{4} (4 x) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

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

Rewrite using the commutative property of multiplication.

$f (x) = - 16 \cdot (4 x^{5} x) - 16 x^{5} \cdot 3 + 24 x^{4} (4 x) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.2

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

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Step 1.6.1.2.1

Move $x$ .

$f (x) = - 16 \cdot (4 (x \cdot x^{5})) - 16 x^{5} \cdot 3 + 24 x^{4} (4 x) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.2.2

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

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Step 1.6.1.2.2.1

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

$f (x) = - 16 \cdot (4 (x \cdot x^{5})) - 16 x^{5} \cdot 3 + 24 x^{4} (4 x) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.2.2.2

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

$f (x) = - 16 \cdot (4 x^{1 + 5}) - 16 x^{5} \cdot 3 + 24 x^{4} (4 x) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.2.3

Add $1$ and $5$ .

$f (x) = - 16 \cdot (4 x^{6}) - 16 x^{5} \cdot 3 + 24 x^{4} (4 x) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.3

Multiply $- 16$ by $4$ .

$f (x) = - 64 x^{6} - 16 x^{5} \cdot 3 + 24 x^{4} (4 x) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.4

Multiply $3$ by $- 16$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 24 x^{4} (4 x) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.5

Rewrite using the commutative property of multiplication.

$f (x) = - 64 x^{6} - 48 x^{5} + 24 \cdot (4 x^{4} x) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.6

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

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Step 1.6.1.6.1

Move $x$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 24 \cdot (4 (x \cdot x^{4})) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.6.2

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

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Step 1.6.1.6.2.1

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

$f (x) = - 64 x^{6} - 48 x^{5} + 24 \cdot (4 (x \cdot x^{4})) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.6.2.2

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

$f (x) = - 64 x^{6} - 48 x^{5} + 24 \cdot (4 x^{1 + 4}) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.6.3

Add $1$ and $4$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 24 \cdot (4 x^{5}) + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.7

Multiply $24$ by $4$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 24 x^{4} \cdot 3 - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.8

Multiply $3$ by $24$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 12 x^{3} (4 x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.9

Rewrite using the commutative property of multiplication.

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 12 \cdot (4 x^{3} x) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.10

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

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Step 1.6.1.10.1

Move $x$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 12 \cdot (4 (x \cdot x^{3})) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.10.2

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

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Step 1.6.1.10.2.1

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

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 12 \cdot (4 (x \cdot x^{3})) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.10.2.2

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

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 12 \cdot (4 x^{1 + 3}) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.10.3

Add $1$ and $3$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 12 \cdot (4 x^{4}) - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.11

Multiply $- 12$ by $4$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 48 x^{4} - 12 x^{3} \cdot 3 + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.12

Multiply $3$ by $- 12$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 48 x^{4} - 36 x^{3} + 2 x^{2} (4 x) + 2 x^{2} \cdot 3$

Step 1.6.1.13

Rewrite using the commutative property of multiplication.

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 48 x^{4} - 36 x^{3} + 2 \cdot (4 x^{2} x) + 2 x^{2} \cdot 3$

Step 1.6.1.14

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

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Step 1.6.1.14.1

Move $x$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 48 x^{4} - 36 x^{3} + 2 \cdot (4 (x \cdot x^{2})) + 2 x^{2} \cdot 3$

Step 1.6.1.14.2

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

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Step 1.6.1.14.2.1

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

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 48 x^{4} - 36 x^{3} + 2 \cdot (4 (x \cdot x^{2})) + 2 x^{2} \cdot 3$

Step 1.6.1.14.2.2

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

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 48 x^{4} - 36 x^{3} + 2 \cdot (4 x^{1 + 2}) + 2 x^{2} \cdot 3$

Step 1.6.1.14.3

Add $1$ and $2$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 48 x^{4} - 36 x^{3} + 2 \cdot (4 x^{3}) + 2 x^{2} \cdot 3$

Step 1.6.1.15

Multiply $2$ by $4$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 48 x^{4} - 36 x^{3} + 8 x^{3} + 2 x^{2} \cdot 3$

Step 1.6.1.16

Multiply $3$ by $2$ .

$f (x) = - 64 x^{6} - 48 x^{5} + 96 x^{5} + 72 x^{4} - 48 x^{4} - 36 x^{3} + 8 x^{3} + 6 x^{2}$

Step 1.6.2

Simplify by adding terms.

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Step 1.6.2.1

Add $- 48 x^{5}$ and $96 x^{5}$ .

$f (x) = - 64 x^{6} + 48 x^{5} + 72 x^{4} - 48 x^{4} - 36 x^{3} + 8 x^{3} + 6 x^{2}$

Step 1.6.2.2

Subtract $48 x^{4}$ from $72 x^{4}$ .

$f (x) = - 64 x^{6} + 48 x^{5} + 24 x^{4} - 36 x^{3} + 8 x^{3} + 6 x^{2}$

Step 1.6.2.3

Add $- 36 x^{3}$ and $8 x^{3}$ .

$f (x) = - 64 x^{6} + 48 x^{5} + 24 x^{4} - 28 x^{3} + 6 x^{2}$

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.

$- 64 x^{6} \to 6$

$48 x^{5} \to 5$

$24 x^{4} \to 4$

$- 28 x^{3} \to 3$

$6 x^{2} \to 2$

Step 2.2

The largest exponent is the degree of the polynomial.

$6$

Step 3

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

$- 64 x^{6}$

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.

$- 64 x^{6}$

Step 4.2

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

$- 64$

Step 5

List the results.

Polynomial Degree: $6$

Leading Term: $- 64 x^{6}$

Leading Coefficient: $- 64$