Question

describe the possible errors made when factoring $25x^2 - 81$ if you get an answer of $(5x + 9)(x - 9)$. then factor correctly.

Explanation:

1. Error Identification: The given answer $(5x + 9)(x - 9)$ fails to follow the difference of squares pattern for $25x^2 - 81$. First, $25x^2$ is the square of $5x$, and $81$ is the square of $9$, so both factors should include $5x$ (not just one). Second, multiplying $(5x + 9)(x - 9)$ gives $5x^2 - 36x - 81$, which does not match the original expression.
2. Correct Factoring: The expression $25x^2 - 81$ is a difference of squares, which factors as $a^2 - b^2 = (a + b)(a - b)$, where $a = 5x$ and $b = 9$.

Answer:

1. Possible Errors:

• Only one factor uses $5x$ instead of both, breaking the difference of squares structure.
• The constant terms in the factors are mismatched, so expanding the incorrect answer does not yield the original expression.

1. Correct Factored Form: $(5x + 9)(5x - 9)$