Algebra Examples

$y = - 2 x (x^{2} - 4) {(x + 1)}^{2} (x - 5)$

Step 1

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

Step 2

If $(x, y)$ exists on the graph, then the graph is symmetric about the:

1. X-Axis if $(x, - y)$ exists on the graph

2. Y-Axis if $(- x, y)$ exists on the graph

3. Origin if $(- x, - y)$ exists on the graph

Step 3

Apply the distributive property.

$y = (- 2 x \cdot x^{2} - 2 x \cdot - 4) {(x + 1)}^{2} (x - 5)$

Step 4

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

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Step 4.1

Move $x^{2}$ .

$y = (- 2 (x^{2} x) - 2 x \cdot - 4) {(x + 1)}^{2} (x - 5)$

Step 4.2

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

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Step 4.2.1

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

$y = (- 2 (x^{2} x^{1}) - 2 x \cdot - 4) {(x + 1)}^{2} (x - 5)$

Step 4.2.2

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

$y = (- 2 x^{2 + 1} - 2 x \cdot - 4) {(x + 1)}^{2} (x - 5)$

Step 4.3

Add $2$ and $1$ .

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

Step 5

Multiply $- 4$ by $- 2$ .

$y = (- 2 x^{3} + 8 x) {(x + 1)}^{2} (x - 5)$

Step 6

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

$y = (- 2 x^{3} + 8 x) ((x + 1) (x + 1)) (x - 5)$

Step 7

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

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Step 7.1

Apply the distributive property.

$y = (- 2 x^{3} + 8 x) (x (x + 1) + 1 (x + 1)) (x - 5)$

Step 7.2

Apply the distributive property.

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $x$ by $x$ .

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

Step 8.1.2

Multiply $x$ by $1$ .

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

Step 8.1.3

Multiply $x$ by $1$ .

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

Step 8.1.4

Multiply $1$ by $1$ .

$y = (- 2 x^{3} + 8 x) (x^{2} + x + x + 1) (x - 5)$

Step 8.2

Add $x$ and $x$ .

$y = (- 2 x^{3} + 8 x) (x^{2} + 2 x + 1) (x - 5)$

Step 9

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

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

Step 10

Simplify each term.

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Step 10.1

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

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Step 10.1.1

Move $x^{2}$ .

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

Step 10.1.2

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

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

Step 10.1.3

Add $2$ and $3$ .

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

Step 10.2

Rewrite using the commutative property of multiplication.

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

Step 10.3

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

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Step 10.3.1

Move $x$ .

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

Step 10.3.2

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

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Step 10.3.2.1

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

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

Step 10.3.2.2

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

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

Step 10.3.3

Add $1$ and $3$ .

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

Step 10.4

Multiply $- 2$ by $2$ .

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

Step 10.5

Multiply $- 2$ by $1$ .

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

Step 10.6

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

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Step 10.6.1

Move $x^{2}$ .

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

Step 10.6.2

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

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Step 10.6.2.1

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

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

Step 10.6.2.2

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

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

Step 10.6.3

Add $2$ and $1$ .

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

Step 10.7

Rewrite using the commutative property of multiplication.

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

Step 10.8

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

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Step 10.8.1

Move $x$ .

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

Step 10.8.2

Multiply $x$ by $x$ .

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

Step 10.9

Multiply $8$ by $2$ .

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

Step 10.10

Multiply $8$ by $1$ .

$y = (- 2 x^{5} - 4 x^{4} - 2 x^{3} + 8 x^{3} + 16 x^{2} + 8 x) (x - 5)$

Step 11

Add $- 2 x^{3}$ and $8 x^{3}$ .

$y = (- 2 x^{5} - 4 x^{4} + 6 x^{3} + 16 x^{2} + 8 x) (x - 5)$

Step 12

Expand $(- 2 x^{5} - 4 x^{4} + 6 x^{3} + 16 x^{2} + 8 x) (x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$y = - 2 x^{5} x - 2 x^{5} \cdot - 5 - 4 x^{4} x - 4 x^{4} \cdot - 5 + 6 x^{3} x + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13

Simplify each term.

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Step 13.1

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

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Step 13.1.1

Move $x$ .

$y = - 2 (x \cdot x^{5}) - 2 x^{5} \cdot - 5 - 4 x^{4} x - 4 x^{4} \cdot - 5 + 6 x^{3} x + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.1.2

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

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Step 13.1.2.1

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

$y = - 2 (x^{1} x^{5}) - 2 x^{5} \cdot - 5 - 4 x^{4} x - 4 x^{4} \cdot - 5 + 6 x^{3} x + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.1.2.2

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

$y = - 2 x^{1 + 5} - 2 x^{5} \cdot - 5 - 4 x^{4} x - 4 x^{4} \cdot - 5 + 6 x^{3} x + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.1.3

Add $1$ and $5$ .

$y = - 2 x^{6} - 2 x^{5} \cdot - 5 - 4 x^{4} x - 4 x^{4} \cdot - 5 + 6 x^{3} x + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.2

Multiply $- 5$ by $- 2$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{4} x - 4 x^{4} \cdot - 5 + 6 x^{3} x + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.3

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

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Step 13.3.1

Move $x$ .

$y = - 2 x^{6} + 10 x^{5} - 4 (x \cdot x^{4}) - 4 x^{4} \cdot - 5 + 6 x^{3} x + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.3.2

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

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Step 13.3.2.1

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

$y = - 2 x^{6} + 10 x^{5} - 4 (x^{1} x^{4}) - 4 x^{4} \cdot - 5 + 6 x^{3} x + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.3.2.2

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

$y = - 2 x^{6} + 10 x^{5} - 4 x^{1 + 4} - 4 x^{4} \cdot - 5 + 6 x^{3} x + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.3.3

Add $1$ and $4$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} - 4 x^{4} \cdot - 5 + 6 x^{3} x + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.4

Multiply $- 5$ by $- 4$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{3} x + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.5

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

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Step 13.5.1

Move $x$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 (x \cdot x^{3}) + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.5.2

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

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Step 13.5.2.1

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

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 (x^{1} x^{3}) + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.5.2.2

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

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{1 + 3} + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.5.3

Add $1$ and $3$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{4} + 6 x^{3} \cdot - 5 + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.6

Multiply $- 5$ by $6$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{4} - 30 x^{3} + 16 x^{2} x + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.7

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

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Step 13.7.1

Move $x$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{4} - 30 x^{3} + 16 (x \cdot x^{2}) + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.7.2

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

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Step 13.7.2.1

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

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{4} - 30 x^{3} + 16 (x^{1} x^{2}) + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.7.2.2

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

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{4} - 30 x^{3} + 16 x^{1 + 2} + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.7.3

Add $1$ and $2$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{4} - 30 x^{3} + 16 x^{3} + 16 x^{2} \cdot - 5 + 8 x \cdot x + 8 x \cdot - 5$

Step 13.8

Multiply $- 5$ by $16$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{4} - 30 x^{3} + 16 x^{3} - 80 x^{2} + 8 x \cdot x + 8 x \cdot - 5$

Step 13.9

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

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Step 13.9.1

Move $x$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{4} - 30 x^{3} + 16 x^{3} - 80 x^{2} + 8 (x \cdot x) + 8 x \cdot - 5$

Step 13.9.2

Multiply $x$ by $x$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{4} - 30 x^{3} + 16 x^{3} - 80 x^{2} + 8 x^{2} + 8 x \cdot - 5$

Step 13.10

Multiply $- 5$ by $8$ .

$y = - 2 x^{6} + 10 x^{5} - 4 x^{5} + 20 x^{4} + 6 x^{4} - 30 x^{3} + 16 x^{3} - 80 x^{2} + 8 x^{2} - 40 x$

Step 14

Subtract $4 x^{5}$ from $10 x^{5}$ .

$y = - 2 x^{6} + 6 x^{5} + 20 x^{4} + 6 x^{4} - 30 x^{3} + 16 x^{3} - 80 x^{2} + 8 x^{2} - 40 x$

Step 15

Add $20 x^{4}$ and $6 x^{4}$ .

$y = - 2 x^{6} + 6 x^{5} + 26 x^{4} - 30 x^{3} + 16 x^{3} - 80 x^{2} + 8 x^{2} - 40 x$

Step 16

Add $- 30 x^{3}$ and $16 x^{3}$ .

$y = - 2 x^{6} + 6 x^{5} + 26 x^{4} - 14 x^{3} - 80 x^{2} + 8 x^{2} - 40 x$

Step 17

Add $- 80 x^{2}$ and $8 x^{2}$ .

$y = - 2 x^{6} + 6 x^{5} + 26 x^{4} - 14 x^{3} - 72 x^{2} - 40 x$

Step 18

Check if the graph is symmetric about the $x$ -axis by plugging in $- y$ for $y$ .

$- y = - 2 x^{6} + 6 x^{5} + 26 x^{4} - 14 x^{3} - 72 x^{2} - 40 x$

Step 19

Since the equation is not identical to the original equation, it is not symmetric to the x-axis.

Not symmetric to the x-axis

Step 20

Check if the graph is symmetric about the $y$ -axis by plugging in $- x$ for $x$ .

$y = - 2 {(- x)}^{6} + 6 {(- x)}^{5} + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 21

Simplify each term.

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Step 21.1

Apply the product rule to $- x$ .

$y = - 2 ({(- 1)}^{6} x^{6}) + 6 {(- x)}^{5} + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 21.2

Raise $- 1$ to the power of $6$ .

$y = - 2 (1 x^{6}) + 6 {(- x)}^{5} + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 21.3

Multiply $x^{6}$ by $1$ .

$y = - 2 x^{6} + 6 {(- x)}^{5} + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 21.4

Apply the product rule to $- x$ .

$y = - 2 x^{6} + 6 ({(- 1)}^{5} x^{5}) + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 21.5

Raise $- 1$ to the power of $5$ .

$y = - 2 x^{6} + 6 (- x^{5}) + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 21.6

Multiply $- 1$ by $6$ .

$y = - 2 x^{6} - 6 x^{5} + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 21.7

Apply the product rule to $- x$ .

$y = - 2 x^{6} - 6 x^{5} + 26 ({(- 1)}^{4} x^{4}) - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 21.8

Raise $- 1$ to the power of $4$ .

$y = - 2 x^{6} - 6 x^{5} + 26 (1 x^{4}) - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 21.9

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

$y = - 2 x^{6} - 6 x^{5} + 26 x^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 21.10

Apply the product rule to $- x$ .

$y = - 2 x^{6} - 6 x^{5} + 26 x^{4} - 14 ({(- 1)}^{3} x^{3}) - 72 {(- x)}^{2} - 40 (- x)$

Step 21.11

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

$y = - 2 x^{6} - 6 x^{5} + 26 x^{4} - 14 (- x^{3}) - 72 {(- x)}^{2} - 40 (- x)$

Step 21.12

Multiply $- 1$ by $- 14$ .

$y = - 2 x^{6} - 6 x^{5} + 26 x^{4} + 14 x^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 21.13

Apply the product rule to $- x$ .

$y = - 2 x^{6} - 6 x^{5} + 26 x^{4} + 14 x^{3} - 72 ({(- 1)}^{2} x^{2}) - 40 (- x)$

Step 21.14

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

$y = - 2 x^{6} - 6 x^{5} + 26 x^{4} + 14 x^{3} - 72 (1 x^{2}) - 40 (- x)$

Step 21.15

Multiply $x^{2}$ by $1$ .

$y = - 2 x^{6} - 6 x^{5} + 26 x^{4} + 14 x^{3} - 72 x^{2} - 40 (- x)$

Step 21.16

Multiply $- 1$ by $- 40$ .

$y = - 2 x^{6} - 6 x^{5} + 26 x^{4} + 14 x^{3} - 72 x^{2} + 40 x$

Step 22

Since the equation is not identical to the original equation, it is not symmetric to the y-axis.

Not symmetric to the y-axis

Step 23

Check if the graph is symmetric about the origin by plugging in $- x$ for $x$ and $- y$ for $y$ .

$- y = - 2 {(- x)}^{6} + 6 {(- x)}^{5} + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 24

Simplify each term.

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Step 24.1

Apply the product rule to $- x$ .

$- y = - 2 ({(- 1)}^{6} x^{6}) + 6 {(- x)}^{5} + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 24.2

Raise $- 1$ to the power of $6$ .

$- y = - 2 (1 x^{6}) + 6 {(- x)}^{5} + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 24.3

Multiply $x^{6}$ by $1$ .

$- y = - 2 x^{6} + 6 {(- x)}^{5} + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 24.4

Apply the product rule to $- x$ .

$- y = - 2 x^{6} + 6 ({(- 1)}^{5} x^{5}) + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 24.5

Raise $- 1$ to the power of $5$ .

$- y = - 2 x^{6} + 6 (- x^{5}) + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 24.6

Multiply $- 1$ by $6$ .

$- y = - 2 x^{6} - 6 x^{5} + 26 {(- x)}^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 24.7

Apply the product rule to $- x$ .

$- y = - 2 x^{6} - 6 x^{5} + 26 ({(- 1)}^{4} x^{4}) - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 24.8

Raise $- 1$ to the power of $4$ .

$- y = - 2 x^{6} - 6 x^{5} + 26 (1 x^{4}) - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 24.9

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

$- y = - 2 x^{6} - 6 x^{5} + 26 x^{4} - 14 {(- x)}^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 24.10

Apply the product rule to $- x$ .

$- y = - 2 x^{6} - 6 x^{5} + 26 x^{4} - 14 ({(- 1)}^{3} x^{3}) - 72 {(- x)}^{2} - 40 (- x)$

Step 24.11

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

$- y = - 2 x^{6} - 6 x^{5} + 26 x^{4} - 14 (- x^{3}) - 72 {(- x)}^{2} - 40 (- x)$

Step 24.12

Multiply $- 1$ by $- 14$ .

$- y = - 2 x^{6} - 6 x^{5} + 26 x^{4} + 14 x^{3} - 72 {(- x)}^{2} - 40 (- x)$

Step 24.13

Apply the product rule to $- x$ .

$- y = - 2 x^{6} - 6 x^{5} + 26 x^{4} + 14 x^{3} - 72 ({(- 1)}^{2} x^{2}) - 40 (- x)$

Step 24.14

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

$- y = - 2 x^{6} - 6 x^{5} + 26 x^{4} + 14 x^{3} - 72 (1 x^{2}) - 40 (- x)$

Step 24.15

Multiply $x^{2}$ by $1$ .

$- y = - 2 x^{6} - 6 x^{5} + 26 x^{4} + 14 x^{3} - 72 x^{2} - 40 (- x)$

Step 24.16

Multiply $- 1$ by $- 40$ .

$- y = - 2 x^{6} - 6 x^{5} + 26 x^{4} + 14 x^{3} - 72 x^{2} + 40 x$

Step 25

Since the equation is not identical to the original equation, it is not symmetric to the origin.

Not symmetric to the origin

Step 26

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Step 27