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Chapter 0: Problem 86

Factor completely, or state that the polynomial is prime. $$x^{2}-10 x+25-36 y^{2}$$

Short Answer

Expert verified
The factored form of the polynomial \(x^{2}-10x+25-36y^{2}\) is \((x - 5 - 6y)(x - 5 + 6y)\).

Step by step solution

01

Identify the Perfect Squares

Look for perfect squares in the polynomial given. Here, \(x^{2}-10x+25\) is the perfect square of \(x-5\) and \(-36y^{2}\) is the perfect square of \(6y\). So rewrite the given expression as: \((x-5)^{2} - (6y)^{2}\)
02

Apply the Difference of Squares Rule

To further factorize, use the difference of squares formula \(a^{2} - b^{2} = (a + b)(a - b)\). The expression \((x-5)^{2} - (6y)^{2}\) can therefore be written as \((x - 5 - 6y)( x - 5 + 6y)\). Expand the terms in the brackets for clarity.
03

Check the Solution

To ensure the solution is correct, expand the factored expression obtained in the previous step. The expression \((x - 5 - 6y)(x - 5+ 6y)\) should simplify back to the given polynomial.

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