Question

a student simplified a rational expression as shown.
\\(\frac{20 - 5x^2}{x^2 + x - 6} = \frac{5(4 - x^2)}{(x - 2)(x + 3)}\\)
\\(= \frac{5(2 - x)(2 + x)}{(x - 2)(x + 3)}\\)
\\(= \frac{5(x + 2)}{(x + 3)}\\)
which of the following mistakes did the student make?
\\(\circ\\) the student did not factor the numerator correctly.
\\(\circ\\) the student could not divide out \\((2 - x)\\) and \\((x - 2)\\) as written.
\\(\circ\\) the student did not factor the denominator correctly
\\(\circ\\) the student needed to simplify further by dividing out \\(x\\).

Explanation:

1. Analyze the numerator and denominator factoring:

• The numerator \(20 - 5x^{2}\) is factored as \(5(4 - x^{2})\) and then further as \(5(2 - x)(2 + x)\), which is correct (using difference of squares \(a^{2}-b^{2}=(a - b)(a + b)\) with \(a = 2\), \(b=x\)).
• The denominator \(x^{2}+x - 6\) is factored as \((x - 2)(x + 3)\), which is correct (since \((x - 2)(x+3)=x^{2}+3x-2x - 6=x^{2}+x - 6\)).

1. Analyze the cancellation:

• The terms \((2 - x)\) and \((x - 2)\) are not the same. In fact, \((2 - x)=-(x - 2)\). For cancellation, we need a common factor. The student tried to cancel \((2 - x)\) and \((x - 2)\) directly, but they are negatives of each other, so they can't be divided out as written.
• The option about not factoring the numerator correctly is wrong because the numerator was factored correctly (factored out 5 first, then used difference of squares).
• The option about not factoring the denominator correctly is wrong as the denominator factoring is correct.
• The option about simplifying further by dividing out \(x\) is wrong because there is no common factor of \(x\) in the numerator and denominator after factoring.

Answer:

The student could not divide out \((2 - x)\) and \((x - 2)\) as written.