Polynomial Factoring Efecca

1. **State the problem:** We need to match each polynomial on the left with its correct factored form on the right. 2. **Recall factoring formulas:** - Difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$ - Perfect square trinomial: $$a^2 - 2ab + b^2 = (a - b)^2$$ 3. **Match $x^2 - 36$:** - Recognize $36 = 6^2$, so $x^2 - 36 = x^2 - 6^2$. - Using difference of squares: $$x^2 - 6^2 = (x - 6)(x + 6)$$ 4. **Match $x^2 - 12x + 36$:** - Note $36 = 6^2$ and $-12x = -2 \times 6 \times x$. - This fits perfect square trinomial: $$x^2 - 2 \times 6 \times x + 6^2 = (x - 6)^2 = (x - 6)(x - 6)$$ 5. **Check remaining factored forms:** - $(x - 9)(x - 4)$ expands to $x^2 - 13x + 36$, which does not match any polynomial on the left. - $(x + 6)(x + 6)$ expands to $x^2 + 12x + 36$, which does not match any polynomial on the left. **Final matches:** - $x^2 - 36$ matches $(x - 6)(x + 6)$ - $x^2 - 12x + 36$ matches $(x - 6)(x - 6)$ **Answer:** - $x^2 - 36 \leftrightarrow (x - 6)(x + 6)$ - $x^2 - 12x + 36 \leftrightarrow (x - 6)(x - 6)$