Trigonometry Examples

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

Step 1

Expand

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

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

Step 2

Simplify each term.

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Step 2.1

Multiply

$x$ by

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

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Step 2.1.1

Multiply

$x$ by

$x^{2}$ .

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Step 2.1.1.1

Raise

$x$ to the power of

$1$ .

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

Step 2.1.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.2

Add

$1$ and

$2$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Multiply

$x$ by

$x$ by adding the exponents.

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Step 2.3.1

Move

$x$ .

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

Step 2.3.2

Multiply

$x$ by

$x$ .

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

Step 2.4

Rewrite using the commutative property of multiplication.

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

Step 2.5

Multiply

$y$ by

$y$ by adding the exponents.

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Step 2.5.1

Move

$y$ .

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

Step 2.5.2

Multiply

$y$ by

$y$ .

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y \cdot y^{2}$

Step 2.6

Multiply

$y$ by

$y^{2}$ by adding the exponents.

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Step 2.6.1

Multiply

$y$ by

$y^{2}$ .

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Step 2.6.1.1

Raise

$y$ to the power of

$1$ .

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y^{1} y^{2}$

Step 2.6.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y^{1 + 2}$

Step 2.6.2

Add

$1$ and

$2$ .

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y^{3}$

Step 3

Combine the opposite terms in

$x^{3} - x^{2} y + x y^{2} + y x^{2} - y^{2} x + y^{3}$ .

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Step 3.1

Reorder the factors in the terms

$- x^{2} y$ and

$y x^{2}$ .

$x^{3} + y^{3} = x^{3} - x^{2} y + x y^{2} + x^{2} y - y^{2} x + y^{3}$

Step 3.2

Add

$- x^{2} y$ and

$x^{2} y$ .

$x^{3} + y^{3} = x^{3} + x y^{2} + 0 - y^{2} x + y^{3}$

Step 3.3

Add

$x^{3} + x y^{2}$ and

$0$ .

$x^{3} + y^{3} = x^{3} + x y^{2} - y^{2} x + y^{3}$

Step 3.4

Reorder the factors in the terms

$x y^{2}$ and

$- y^{2} x$ .

$x^{3} + y^{3} = x^{3} + y^{2} x - y^{2} x + y^{3}$

Step 3.5

Subtract

$y^{2} x$ from

$y^{2} x$ .

$x^{3} + y^{3} = x^{3} + 0 + y^{3}$

Step 3.6

Add

$x^{3}$ and

$0$ .

$x^{3} + y^{3} = x^{3} + y^{3}$

Step 4

Since the two sides have been shown to be equivalent, the equation is an identity.

$x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})$ is an identity.