Algebra Examples

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

Step 1

Regroup terms.

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

Step 2

Factor $x$ out of $a^{2} x + 3 a x + 2 x$ .

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Step 2.1

Factor $x$ out of $a^{2} x$ .

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

Step 2.2

Factor $x$ out of $3 a x$ .

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

Step 2.3

Factor $x$ out of $2 x$ .

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

Step 2.4

Factor $x$ out of $x a^{2} + x (3 a)$ .

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

Step 2.5

Factor $x$ out of $x (a^{2} + 3 a) + x \cdot 2$ .

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

Step 3

Factor.

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Step 3.1

Factor $a^{2} + 3 a + 2$ using the AC method.

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Step 3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 3.1.2

Write the factored form using these integers.

$x ((a + 1) (a + 2)) - a^{2} y - 3 a y - 2 y$

Step 3.2

Remove unnecessary parentheses.

$x (a + 1) (a + 2) - a^{2} y - 3 a y - 2 y$

Step 4

Factor $y$ out of $- a^{2} y - 3 a y - 2 y$ .

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Step 4.1

Factor $y$ out of $- a^{2} y$ .

$x (a + 1) (a + 2) + y (- a^{2}) - 3 a y - 2 y$

Step 4.2

Factor $y$ out of $- 3 a y$ .

$x (a + 1) (a + 2) + y (- a^{2}) + y (- 3 a) - 2 y$

Step 4.3

Factor $y$ out of $- 2 y$ .

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

Step 4.4

Factor $y$ out of $y (- a^{2}) + y (- 3 a)$ .

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

Step 4.5

Factor $y$ out of $y (- a^{2} - 3 a) + y \cdot - 2$ .

$x (a + 1) (a + 2) + y (- a^{2} - 3 a - 2)$

Step 5

Factor.

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Step 5.1

Factor by grouping.

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Step 5.1.1

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 $a \cdot c = - 1 \cdot - 2 = 2$ and whose sum is $b = - 3$ .

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Step 5.1.1.1

Factor $- 3$ out of $- 3 a$ .

$x (a + 1) (a + 2) + y (- a^{2} - 3 (a) - 2)$

Step 5.1.1.2

Rewrite $- 3$ as $- 1$ plus $- 2$

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

Step 5.1.1.3

Apply the distributive property.

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

Step 5.1.2

Factor out the greatest common factor from each group.

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Step 5.1.2.1

Group the first two terms and the last two terms.

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

Step 5.1.2.2

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

$x (a + 1) (a + 2) + y (a (- a - 1) + 2 (- a - 1))$

Step 5.1.3

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

$x (a + 1) (a + 2) + y ((- a - 1) (a + 2))$

Step 5.2

Remove unnecessary parentheses.

$x (a + 1) (a + 2) + y (- a - 1) (a + 2)$

Step 6

Factor $a + 2$ out of $x (a + 1) (a + 2) + y (- a - 1) (a + 2)$ .

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Step 6.1

Factor $a + 2$ out of $x (a + 1) (a + 2)$ .

$(a + 2) (x (a + 1)) + y (- a - 1) (a + 2)$

Step 6.2

Factor $a + 2$ out of $y (- a - 1) (a + 2)$ .

$(a + 2) (x (a + 1)) + (a + 2) (y (- a - 1))$

Step 6.3

Factor $a + 2$ out of $(a + 2) (x (a + 1)) + (a + 2) (y (- a - 1))$ .

$(a + 2) (x (a + 1) + y (- a - 1))$

Step 7

Apply the distributive property.

$(a + 2) (x a + x \cdot 1 + y (- a - 1))$

Step 8

Multiply $x$ by $1$ .

$(a + 2) (x a + x + y (- a - 1))$

Step 9

Apply the distributive property.

$(a + 2) (x a + x + y (- a) + y \cdot - 1)$

Step 10

Rewrite using the commutative property of multiplication.

$(a + 2) (x a + x - y a + y \cdot - 1)$

Step 11

Move $- 1$ to the left of $y$ .

$(a + 2) (x a + x - y a - 1 y)$

Step 12

Rewrite $- 1 y$ as $- y$ .

$(a + 2) (x a + x - y a - y)$

Step 13

Factor.

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Step 13.1

Rewrite $x a + x - y a - y$ in a factored form.

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Step 13.1.1

Factor out the greatest common factor from each group.

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Step 13.1.1.1

Group the first two terms and the last two terms.

$(a + 2) ((x a + x) - y a - y)$

Step 13.1.1.2

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

$(a + 2) (x (a + 1) - y (a + 1))$

Step 13.1.2

Factor the polynomial by factoring out the greatest common factor, $a + 1$ .

$(a + 2) ((a + 1) (x - y))$

Step 13.2

Remove unnecessary parentheses.

$(a + 2) (a + 1) (x - y)$