Problem 86 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The complete factorization of the polynomial \(x^{2}-10x+25-36y^{2}\) is \((x - 5 - 6y)(x - 5+ 6y)\).

Step 1: Identifying polynomial types

Check if the given polynomial can be identified as a difference of squares, difference of cubes, sum of cubes, or a perfect square trinomial. In this case, the polynomial \(x^{2}-10x+25-36y^{2}\) is a difference between a perfect square trinomial and a perfect square, which is a type of difference of squares.

Step 2: Factoring the perfect square trinomial and perfect squares

Factor the perfect square trinomial \(x^{2}-10x+25\) to be \((x-5)^2\). Also, write \(36y^2\) as \((6y)^2\). So, the polynomial can be rewritten as \((x-5)^2 - (6y)^2\).

Step 3: Use difference of squares pattern

Use the difference of squares pattern which can be expressed as \(a^2 - b^2 = (a - b)(a + b)\). Applying this to \((x-5)^2 - (6y)^2\), we get \((x-5 - 6y)(x-5+ 6y)\).

Step 4: Finalize the factors

Write the final factors in the standard form. The fully factored version of the given polynomial is \((x - 5 - 6y)(x - 5+ 6y)\).