Problem 86 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The completely factored form of the given polynomial is \((x-5 + 6y)(x-5 - 6y)\)

Step 1: Rearrange and Identify the Squares

Rearrange the equation as \((x^{2}-10x+25) - 36y^{2}\), hence identifying \(a^2\) as \((x^{2}-10x+25)\) and \(b^2\) as \(36y^{2}\)

Step 2: Identify the terms as Perfect Squares and Rewrite

The first expression \((x^{2}-10x+25)\) can be rewritten as \((x-5)^2\) because \(a^2 - 2ab + b^2 = (a-b)^2\). Also \(36y^{2}\) can be rewritten as \((6y)^2\)

Step 3: Apply the 'Difference of Squares' formula

Applying the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), you obtain \((x-5 + 6y)(x-5 - 6y)\)