Write the polynomial as an equation.

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

If <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mstyle></math> exists on the graph, then the graph is symmetric about the:

1. X-Axis if <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mo>-</mo><mi>y</mi><mo>)</mo></mrow></mstyle></math> exists on the graph

2. Y-Axis if <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mstyle></math> exists on the graph

3. Origin if <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mi>x</mi><mo>,</mo><mo>-</mo><mi>y</mi><mo>)</mo></mrow></mstyle></math> exists on the graph

Check if the graph is symmetric about the x-axis by plugging in <math><mstyle displaystyle="true"><mo>-</mo><mi>y</mi></mstyle></math> for <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> .

Remove parentheses.

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 <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

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 <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mo>-</mo><mi>y</mi></mstyle></math> for <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> .

Simplify each term.

Apply the product rule to <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Remove parentheses.

Simplify each term.

Apply the product rule to <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mrow><mo>(</mo><mo>-</mo><mi>y</mi><mo>)</mo></mrow></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mo>-</mo><mrow><mo>(</mo><mn>3</mn><msup><mrow><mo>(</mo><mo>-</mo><mi>x</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mo>-</mo><mi>x</mi><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mstyle></math> .

Simplify each term.

Apply the product rule to <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Apply the product rule to <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> .

Simplify by multiplying through.

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mo>-</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

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

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