Factor each polynomial. See Examples 5 and 6. $y^2-x^2+12x-36$

Verified step by step guidance

1. Identify the polynomial to factor: \(y^2 - x^2 + 12x - 36\).
2. Group the terms to make factoring easier: \(y^2 - (x^2 - 12x + 36)\).
3. Recognize that \(x^2 - 12x + 36\) is a perfect square trinomial, since \$36 = 6^2$ and \(-12x = -2 \cdot 6 \cdot x\).
4. Rewrite the expression using the perfect square: \(y^2 - (x - 6)^2\).
5. Apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\), where \(a = y\) and \(b = (x - 6)\), to factor as \((y - (x - 6))(y + (x - 6))\).

Key Concepts

Polynomial Factoring

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This process helps simplify expressions and solve equations. Common methods include factoring out the greatest common factor, grouping, and special products like difference of squares.

Difference of Squares

The difference of squares is a special factoring pattern where an expression of the form a² - b² factors into (a - b)(a + b). Recognizing this pattern allows quick factoring of certain quadratic expressions, which is essential for simplifying or solving polynomial equations.

Completing the Square

Completing the square transforms a quadratic expression into a perfect square trinomial plus or minus a constant. This technique is useful for factoring quadratics that are not easily factorable by inspection, and it helps identify patterns like difference of squares after rearrangement.