Question

1. error analysis describe and correct the error a student made in factoring $x^2 - 36$.

use the perfect-square trinomial pattern to factor $x^2 - 36$ because both terms are perfect squares.
$x^2 - 36 = (x - 6)(x - 6)$

Explanation:

Step1: Identify the error type

$x^2 - 36$ is a difference of squares, not a perfect-square trinomial (which has 3 terms: $a^2\pm2ab+b^2$). The student misapplied the perfect-square pattern to a binomial difference of squares.

Step2: Apply difference of squares rule

The formula for factoring a difference of squares is $a^2 - b^2=(a+b)(a-b)$. Here, $a=x$, $b=6$ since $x^2=x^2$ and $36=6^2$.
<Expression>
$x^2 - 36 = x^2 - 6^2=(x+6)(x-6)$
</Expression>

Answer:

The student incorrectly used the perfect-square trinomial factoring pattern on a binomial difference of squares. $x^2 - 36$ has only two terms, so it cannot be a perfect-square trinomial. The correct factorization is $\boldsymbol{(x+6)(x-6)}$.