Algebra Examples

$32 x (x - y) - 17 y (y - x)$

Step 1

Apply the distributive property.

$32 x \cdot x + 32 x (- y) - 17 y (y - x)$

Step 2

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

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Step 2.1

Move $x$ .

$32 (x \cdot x) + 32 x (- y) - 17 y (y - x)$

Step 2.2

Multiply $x$ by $x$ .

$32 x^{2} + 32 x (- y) - 17 y (y - x)$

Step 3

Rewrite using the commutative property of multiplication.

$32 x^{2} + 32 \cdot - 1 x y - 17 y (y - x)$

Step 4

Multiply $32$ by $- 1$ .

$32 x^{2} - 32 x y - 17 y (y - x)$

Step 5

Apply the distributive property.

$32 x^{2} - 32 x y - 17 y \cdot y - 17 y (- x)$

Step 6

Multiply $y$ by $y$ by adding the exponents.

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Step 6.1

Move $y$ .

$32 x^{2} - 32 x y - 17 (y \cdot y) - 17 y (- x)$

Step 6.2

Multiply $y$ by $y$ .

$32 x^{2} - 32 x y - 17 y^{2} - 17 y (- x)$

Step 7

Rewrite using the commutative property of multiplication.

$32 x^{2} - 32 x y - 17 y^{2} - 17 \cdot - 1 y x$

Step 8

Multiply $- 17$ by $- 1$ .

$32 x^{2} - 32 x y - 17 y^{2} + 17 y x$

Step 9

Add $- 32 x y$ and $17 y x$ .

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Step 9.1

Move $y$ .

$32 x^{2} - 17 y^{2} - 32 x y + 17 x y$

Step 9.2

Add $- 32 x y$ and $17 x y$ .

$32 x^{2} - 17 y^{2} - 15 x y$

Step 10

Factor by grouping.

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Step 10.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 32 \cdot - 17 = - 544$ and whose sum is $b = - 15$ .

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Step 10.1.1

Reorder $- 17 y^{2}$ and $- 15 x y$ .

$32 x^{2} - 15 x y - 17 y^{2}$

Step 10.1.2

Factor $- 15$ out of $- 15 x y$ .

$32 x^{2} - 15 (x y) - 17 y^{2}$

Step 10.1.3

Rewrite $- 15$ as $17$ plus $- 32$

$32 x^{2} + (17 - 32) (x y) - 17 y^{2}$

Step 10.1.4

Apply the distributive property.

$32 x^{2} + 17 (x y) - 32 (x y) - 17 y^{2}$

Step 10.1.5

Remove unnecessary parentheses.

$32 x^{2} + 17 x y - 32 (x y) - 17 y^{2}$

Step 10.1.6

Remove unnecessary parentheses.

$32 x^{2} + 17 x y - 32 x y - 17 y^{2}$

Step 10.2

Factor out the greatest common factor from each group.

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Step 10.2.1

Group the first two terms and the last two terms.

$(32 x^{2} + 17 x y) - 32 x y - 17 y^{2}$

Step 10.2.2

Factor out the greatest common factor (GCF) from each group.

$x (32 x + 17 y) - y (32 x + 17 y)$

Step 10.3

Factor the polynomial by factoring out the greatest common factor, $32 x + 17 y$ .

$(32 x + 17 y) (x - y)$