Algebra Examples

$1 - x y - 20 x^{2} y^{2}$

Step 1

Factor $- 1$ out of $1 - x y - 20 x^{2} y^{2}$ .

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Step 1.1

Reorder the expression.

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Step 1.1.1

Move $1$ .

$- x y - 20 x^{2} y^{2} + 1$

Step 1.1.2

Reorder $- x y$ and $- 20 x^{2} y^{2}$ .

$- 20 x^{2} y^{2} - x y + 1$

Step 1.2

Factor $- 1$ out of $- 20 x^{2} y^{2}$ .

$- (20 x^{2} y^{2}) - x y + 1$

Step 1.3

Factor $- 1$ out of $- x y$ .

$- (20 x^{2} y^{2}) - (x y) + 1$

Step 1.4

Rewrite $1$ as $- 1 (- 1)$ .

$- (20 x^{2} y^{2}) - (x y) - 1 (- 1)$

Step 1.5

Factor $- 1$ out of $- (20 x^{2} y^{2}) - (x y)$ .

$- (20 x^{2} y^{2} + x y) - 1 (- 1)$

Step 1.6

Factor $- 1$ out of $- (20 x^{2} y^{2} + x y) - 1 (- 1)$ .

$- (20 x^{2} y^{2} + x y - 1)$

Step 2

Rewrite $x^{2} y^{2}$ as ${(x y)}^{2}$ .

$- (20 {(x y)}^{2} + x y - 1)$

Step 3

Let $u = x y$ . Substitute $u$ for all occurrences of $x y$ .

$- (20 u^{2} + u - 1)$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 20 \cdot - 1 = - 20$ and whose sum is $b = 1$ .

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Step 4.1.1

Multiply by $1$ .

$- (20 u^{2} + 1 u - 1)$

Step 4.1.2

Rewrite $1$ as $- 4$ plus $5$

$- (20 u^{2} + (- 4 + 5) u - 1)$

Step 4.1.3

Apply the distributive property.

$- (20 u^{2} - 4 u + 5 u - 1)$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$- ((20 u^{2} - 4 u) + 5 u - 1)$

Step 4.2.2

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

$- (4 u (5 u - 1) + 1 (5 u - 1))$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $5 u - 1$ .

$- ((5 u - 1) (4 u + 1))$

Step 5

Replace all occurrences of $u$ with $x y$ .

$- ((5 (x y) - 1) (4 (x y) + 1))$

Step 6

Factor.

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Step 6.1

Simplify.

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Step 6.1.1

Remove parentheses.

$- ((5 x y - 1) (4 (x y) + 1))$

Step 6.1.2

Remove parentheses.

$- ((5 x y - 1) (4 x y + 1))$

Step 6.2

Remove unnecessary parentheses.

$- (5 x y - 1) (4 x y + 1)$