Find the Symmetry x^4+10x^3+25x^2

$x^{4} + 10 x^{3} + 25 x^{2}$

Write $x^{4} + 10 x^{3} + 25 x^{2}$ as an equation.

$y = x^{4} + 10 x^{3} + 25 x^{2}$

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

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

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

$- y = x^{4} + 10 x^{3} + 25 x^{2}$

Simplify the left side.

Remove parentheses.

$- y = x^{4} + 10 x^{3} + 25 x^{2}$

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

Not symmetric to the x-axis

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

$y = {(- x)}^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2}$

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

$y = {(- 1)}^{4} x^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2}$

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

$y = 1 x^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2}$

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

$y = x^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2}$

Apply the product rule to $- x$ .

$y = x^{4} + 10 ({(- 1)}^{3} x^{3}) + 25 {(- x)}^{2}$

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

$y = x^{4} + 10 (- x^{3}) + 25 {(- x)}^{2}$

Multiply $- 1$ by $10$ .

$y = x^{4} - 10 x^{3} + 25 {(- x)}^{2}$

Apply the product rule to $- x$ .

$y = x^{4} - 10 x^{3} + 25 ({(- 1)}^{2} x^{2})$

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

$y = x^{4} - 10 x^{3} + 25 (1 x^{2})$

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

$y = x^{4} - 10 x^{3} + 25 x^{2}$

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

Not symmetric to the y-axis

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

$- y = {(- x)}^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2}$

Solve for $y$ .

Simplify the left side.

Remove parentheses.

$- y = {(- x)}^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2}$

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

$- y = {(- 1)}^{4} x^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2}$

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

$- y = 1 x^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2}$

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

$- y = x^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2}$

Apply the product rule to $- x$ .

$- y = x^{4} + 10 ({(- 1)}^{3} x^{3}) + 25 {(- x)}^{2}$

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

$- y = x^{4} + 10 (- x^{3}) + 25 {(- x)}^{2}$

Multiply $- 1$ by $10$ .

$- y = x^{4} - 10 x^{3} + 25 {(- x)}^{2}$

Apply the product rule to $- x$ .

$- y = x^{4} - 10 x^{3} + 25 ({(- 1)}^{2} x^{2})$

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

$- y = x^{4} - 10 x^{3} + 25 (1 x^{2})$

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

$- y = x^{4} - 10 x^{3} + 25 x^{2}$

Simplify the left side.

Multiply $- (- y)$ .

Multiply $- 1$ by $- 1$ .

$1 y = - ({(- x)}^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2})$

Multiply $y$ by $1$ .

$y = - ({(- x)}^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2})$

Simplify the right side.

Simplify $- ({(- x)}^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2})$ .

Simplify each term.

Apply the product rule to $- x$ .

$y = - ({(- 1)}^{4} x^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2})$

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

$y = - (1 x^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2})$

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

$y = - (x^{4} + 10 {(- x)}^{3} + 25 {(- x)}^{2})$

Apply the product rule to $- x$ .

$y = - (x^{4} + 10 ({(- 1)}^{3} x^{3}) + 25 {(- x)}^{2})$

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

$y = - (x^{4} + 10 (- x^{3}) + 25 {(- x)}^{2})$

Multiply $- 1$ by $10$ .

$y = - (x^{4} - 10 x^{3} + 25 {(- x)}^{2})$

Apply the product rule to $- x$ .

$y = - (x^{4} - 10 x^{3} + 25 ({(- 1)}^{2} x^{2}))$

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

$y = - (x^{4} - 10 x^{3} + 25 (1 x^{2}))$

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

$y = - (x^{4} - 10 x^{3} + 25 x^{2})$

Apply the distributive property.

$y = - x^{4} - (- 10 x^{3}) - (25 x^{2})$

Simplify.

Multiply $- 10$ by $- 1$ .

$y = - x^{4} + 10 x^{3} - (25 x^{2})$

Multiply $25$ by $- 1$ .

$y = - x^{4} + 10 x^{3} - 25 x^{2}$

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

Not symmetric to the origin

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Find the Symmetry x^4+10x^3+25x^2

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