Trigonometry Examples

$10 x^{2} y^{2} + 3 x y - 18$

Step 1

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

$10 {(x y)}^{2} + 3 (x y) - 18$

Step 2

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

$10 u^{2} + 3 u - 18$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + i x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 10 \cdot - 18 = - 180$ and whose sum is $i = 3$ .

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Step 3.1.1

Factor $3$ out of $3 u$ .

$10 u^{2} + 3 (u) - 18$

Step 3.1.2

Rewrite $3$ as $- 12$ plus $15$

$10 u^{2} + (- 12 + 15) u - 18$

Step 3.1.3

Apply the distributive property.

$10 u^{2} - 12 u + 15 u - 18$

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.

$(10 u^{2} - 12 u) + 15 u - 18$

Step 3.2.2

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

$2 u (5 u - 6) + 3 (5 u - 6)$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $5 u - 6$ .

$(5 u - 6) (2 u + 3)$

Step 4

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

$(5 (x y) - 6) (2 (x y) + 3)$

Step 5

Simplify.

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Step 5.1

Remove parentheses.

$(5 x y - 6) (2 (x y) + 3)$

Step 5.2

Remove parentheses.

$(5 x y - 6) (2 x y + 3)$