Finite Math Examples

$25 y^{2 m} - (x^{2 n} - 2 x^{n} + 1)$

Step 1

Rewrite $25 y^{2 m}$ as ${(5 y^{m})}^{2}$ .

${(5 y^{m})}^{2} - (x^{2 n} - 2 x^{n} + 1)$

Step 2

Rewrite $x^{2 n} - 2 x^{n} + 1$ as ${(x^{n} - 1)}^{2}$ .

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Step 2.1

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

${(5 y^{m})}^{2} - ({(x^{n})}^{2} - 2 x^{n} + 1)$

Step 2.2

Rewrite $1$ as $1^{2}$ .

${(5 y^{m})}^{2} - ({(x^{n})}^{2} - 2 x^{n} + 1^{2})$

Step 2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x^{n} = 2 \cdot x^{n} \cdot 1$

Step 2.4

Rewrite the polynomial.

${(5 y^{m})}^{2} - ({(x^{n})}^{2} - 2 \cdot x^{n} \cdot 1 + 1^{2})$

Step 2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x^{n}$ and $b = 1$ .

${(5 y^{m})}^{2} - {(x^{n} - 1)}^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 y^{m}$ and $b = x^{n} - 1$ .

$(5 y^{m} + x^{n} - 1) (5 y^{m} - (x^{n} - 1))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(5 y^{m} + x^{n} - 1) (5 y^{m} - x^{n} - - 1)$

Step 4.2

Multiply $- 1$ by $- 1$ .

$(5 y^{m} + x^{n} - 1) (5 y^{m} - x^{n} + 1)$