Algebra Examples

$x^{2} {(x - 4)}^{3} (x + 5)$

Step 1

Write $x^{2} {(x - 4)}^{3} (x + 5)$ as an equation.

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

Step 2

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

Step 3

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 4

Use the Binomial Theorem.

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

Step 5

Simplify each term.

Tap for more steps...

Step 5.1

Multiply $- 4$ by $3$ .

$y = x^{2} (x^{3} - 12 x^{2} + 3 x {(- 4)}^{2} + {(- 4)}^{3}) (x + 5)$

Step 5.2

Raise $- 4$ to the power of $2$ .

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

Step 5.3

Multiply $16$ by $3$ .

$y = x^{2} (x^{3} - 12 x^{2} + 48 x + {(- 4)}^{3}) (x + 5)$

Step 5.4

Raise $- 4$ to the power of $3$ .

$y = x^{2} (x^{3} - 12 x^{2} + 48 x - 64) (x + 5)$

Step 6

Apply the distributive property.

$y = (x^{2} x^{3} + x^{2} (- 12 x^{2}) + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Step 7

Simplify.

Tap for more steps...

Step 7.1

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

Tap for more steps...

Step 7.1.1

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

$y = (x^{2 + 3} + x^{2} (- 12 x^{2}) + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Step 7.1.2

Add $2$ and $3$ .

$y = (x^{5} + x^{2} (- 12 x^{2}) + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Step 7.2

Rewrite using the commutative property of multiplication.

$y = (x^{5} - 12 x^{2} x^{2} + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Step 7.3

Rewrite using the commutative property of multiplication.

$y = (x^{5} - 12 x^{2} x^{2} + 48 x^{2} x + x^{2} \cdot - 64) (x + 5)$

Step 7.4

Move $- 64$ to the left of $x^{2}$ .

$y = (x^{5} - 12 x^{2} x^{2} + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

Step 8

Simplify each term.

Tap for more steps...

Step 8.1

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

Tap for more steps...

Step 8.1.1

Move $x^{2}$ .

$y = (x^{5} - 12 (x^{2} x^{2}) + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

Step 8.1.2

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

$y = (x^{5} - 12 x^{2 + 2} + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

Step 8.1.3

Add $2$ and $2$ .

$y = (x^{5} - 12 x^{4} + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

Step 8.2

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

Tap for more steps...

Step 8.2.1

Move $x$ .

$y = (x^{5} - 12 x^{4} + 48 (x \cdot x^{2}) - 64 \cdot x^{2}) (x + 5)$

Step 8.2.2

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

Tap for more steps...

Step 8.2.2.1

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

$y = (x^{5} - 12 x^{4} + 48 (x^{1} x^{2}) - 64 \cdot x^{2}) (x + 5)$

Step 8.2.2.2

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

$y = (x^{5} - 12 x^{4} + 48 x^{1 + 2} - 64 \cdot x^{2}) (x + 5)$

Step 8.2.3

Add $1$ and $2$ .

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

$y = (x^{5} - 12 x^{4} + 48 x^{3} - 64 x^{2}) (x + 5)$

Step 9

Expand $(x^{5} - 12 x^{4} + 48 x^{3} - 64 x^{2}) (x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{5} x + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10

Simplify each term.

Tap for more steps...

Step 10.1

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

Tap for more steps...

Step 10.1.1

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

Tap for more steps...

Step 10.1.1.1

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

$y = x^{5} x^{1} + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.1.1.2

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

$y = x^{5 + 1} + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.1.2

Add $5$ and $1$ .

$y = x^{6} + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.2

Move $5$ to the left of $x^{5}$ .

$y = x^{6} + 5 \cdot x^{5} - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.3

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

Tap for more steps...

Step 10.3.1

Move $x$ .

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

Step 10.3.2

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

Tap for more steps...

Step 10.3.2.1

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

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

Step 10.3.2.2

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

$y = x^{6} + 5 x^{5} - 12 x^{1 + 4} - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.3.3

Add $1$ and $4$ .

$y = x^{6} + 5 x^{5} - 12 x^{5} - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.4

Multiply $5$ by $- 12$ .

$y = x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.5

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

Tap for more steps...

Step 10.5.1

Move $x$ .

$y = x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 (x \cdot x^{3}) + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.5.2

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

Tap for more steps...

Step 10.5.2.1

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

$y = x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 (x^{1} x^{3}) + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.5.2.2

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

$y = x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{1 + 3} + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.5.3

Add $1$ and $3$ .

$y = x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.6

Multiply $5$ by $48$ .

$y = x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{2} x - 64 x^{2} \cdot 5$

Step 10.7

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

Tap for more steps...

Step 10.7.1

Move $x$ .

$y = x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 (x \cdot x^{2}) - 64 x^{2} \cdot 5$

Step 10.7.2

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

Tap for more steps...

Step 10.7.2.1

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

$y = x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 (x^{1} x^{2}) - 64 x^{2} \cdot 5$

Step 10.7.2.2

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

$y = x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{1 + 2} - 64 x^{2} \cdot 5$

Step 10.7.3

Add $1$ and $2$ .

$y = x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{3} - 64 x^{2} \cdot 5$

Step 10.8

Multiply $5$ by $- 64$ .

$y = x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{3} - 320 x^{2}$

Step 11

Subtract $12 x^{5}$ from $5 x^{5}$ .

$y = x^{6} - 7 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{3} - 320 x^{2}$

Step 12

Add $- 60 x^{4}$ and $48 x^{4}$ .

$y = x^{6} - 7 x^{5} - 12 x^{4} + 240 x^{3} - 64 x^{3} - 320 x^{2}$

Step 13

Subtract $64 x^{3}$ from $240 x^{3}$ .

$y = x^{6} - 7 x^{5} - 12 x^{4} + 176 x^{3} - 320 x^{2}$

Step 14

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

$- y = x^{6} - 7 x^{5} - 12 x^{4} + 176 x^{3} - 320 x^{2}$

Step 15

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 16

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

$y = {(- x)}^{6} - 7 {(- x)}^{5} - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 17

Simplify each term.

Tap for more steps...

Step 17.1

Apply the product rule to $- x$ .

$y = {(- 1)}^{6} x^{6} - 7 {(- x)}^{5} - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 17.2

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

$y = 1 x^{6} - 7 {(- x)}^{5} - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 17.3

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

$y = x^{6} - 7 {(- x)}^{5} - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 17.4

Apply the product rule to $- x$ .

$y = x^{6} - 7 ({(- 1)}^{5} x^{5}) - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 17.5

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

$y = x^{6} - 7 (- x^{5}) - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 17.6

Multiply $- 1$ by $- 7$ .

$y = x^{6} + 7 x^{5} - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 17.7

Apply the product rule to $- x$ .

$y = x^{6} + 7 x^{5} - 12 ({(- 1)}^{4} x^{4}) + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 17.8

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

$y = x^{6} + 7 x^{5} - 12 (1 x^{4}) + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 17.9

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

$y = x^{6} + 7 x^{5} - 12 x^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 17.10

Apply the product rule to $- x$ .

$y = x^{6} + 7 x^{5} - 12 x^{4} + 176 ({(- 1)}^{3} x^{3}) - 320 {(- x)}^{2}$

Step 17.11

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

$y = x^{6} + 7 x^{5} - 12 x^{4} + 176 (- x^{3}) - 320 {(- x)}^{2}$

Step 17.12

Multiply $- 1$ by $176$ .

$y = x^{6} + 7 x^{5} - 12 x^{4} - 176 x^{3} - 320 {(- x)}^{2}$

Step 17.13

Apply the product rule to $- x$ .

$y = x^{6} + 7 x^{5} - 12 x^{4} - 176 x^{3} - 320 ({(- 1)}^{2} x^{2})$

Step 17.14

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

$y = x^{6} + 7 x^{5} - 12 x^{4} - 176 x^{3} - 320 (1 x^{2})$

Step 17.15

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

$y = x^{6} + 7 x^{5} - 12 x^{4} - 176 x^{3} - 320 x^{2}$

Step 18

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 19

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

$- y = {(- x)}^{6} - 7 {(- x)}^{5} - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 20

Simplify each term.

Tap for more steps...

Step 20.1

Apply the product rule to $- x$ .

$- y = {(- 1)}^{6} x^{6} - 7 {(- x)}^{5} - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 20.2

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

$- y = 1 x^{6} - 7 {(- x)}^{5} - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 20.3

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

$- y = x^{6} - 7 {(- x)}^{5} - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 20.4

Apply the product rule to $- x$ .

$- y = x^{6} - 7 ({(- 1)}^{5} x^{5}) - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 20.5

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

$- y = x^{6} - 7 (- x^{5}) - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 20.6

Multiply $- 1$ by $- 7$ .

$- y = x^{6} + 7 x^{5} - 12 {(- x)}^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 20.7

Apply the product rule to $- x$ .

$- y = x^{6} + 7 x^{5} - 12 ({(- 1)}^{4} x^{4}) + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 20.8

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

$- y = x^{6} + 7 x^{5} - 12 (1 x^{4}) + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 20.9

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

$- y = x^{6} + 7 x^{5} - 12 x^{4} + 176 {(- x)}^{3} - 320 {(- x)}^{2}$

Step 20.10

Apply the product rule to $- x$ .

$- y = x^{6} + 7 x^{5} - 12 x^{4} + 176 ({(- 1)}^{3} x^{3}) - 320 {(- x)}^{2}$

Step 20.11

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

$- y = x^{6} + 7 x^{5} - 12 x^{4} + 176 (- x^{3}) - 320 {(- x)}^{2}$

Step 20.12

Multiply $- 1$ by $176$ .

$- y = x^{6} + 7 x^{5} - 12 x^{4} - 176 x^{3} - 320 {(- x)}^{2}$

Step 20.13

Apply the product rule to $- x$ .

$- y = x^{6} + 7 x^{5} - 12 x^{4} - 176 x^{3} - 320 ({(- 1)}^{2} x^{2})$

Step 20.14

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

$- y = x^{6} + 7 x^{5} - 12 x^{4} - 176 x^{3} - 320 (1 x^{2})$

Step 20.15

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

$- y = x^{6} + 7 x^{5} - 12 x^{4} - 176 x^{3} - 320 x^{2}$

Step 21

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

Not symmetric to the origin

Step 22

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Step 23