Pre-Algebra Examples

$14 x^{2} y^{2} + 19 x y - 3$

Step 1

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

$14 {(x y)}^{2} + 19 (x y) - 3$

Step 2

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

$14 u^{2} + 19 u - 3$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 14 \cdot - 3 = - 42$ and whose sum is $b = 19$ .

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Step 3.1.1

Factor $19$ out of $19 u$ .

$14 u^{2} + 19 (u) - 3$

Step 3.1.2

Rewrite $19$ as $- 2$ plus $21$

$14 u^{2} + (- 2 + 21) u - 3$

Step 3.1.3

Apply the distributive property.

$14 u^{2} - 2 u + 21 u - 3$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(14 u^{2} - 2 u) + 21 u - 3$

Step 3.2.2

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

$2 u (7 u - 1) + 3 (7 u - 1)$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $7 u - 1$ .

$(7 u - 1) (2 u + 3)$

Step 4

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

$(7 (x y) - 1) (2 (x y) + 3)$

Step 5

Simplify.

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Step 5.1

Remove parentheses.

$(7 x y - 1) (2 (x y) + 3)$

Step 5.2

Remove parentheses.

$(7 x y - 1) (2 x y + 3)$