SAT Polynomials

Correct Answer: C

Solution 1:

Tip: $a^2 - b^2 = (a-b)(a+b)$
Using the difference of squares formula, we obtain $x = (k-2)(k+2) = k^2 - 2^2 = k^2 - 4$ .
Subtracting 4 from both sides, we get $x- 4 = (k^2 -4 ) - 4 = k^2 - 8$ .

Solution 2:

FOIL-ing the right side, we obtain

$\begin{aligned} x &= (k-2)(k+2) \\ & = k\times k - 2 \times k + k \times 2 - 2 \times 2\\ & = k^2 - 2k + 2k - 4 \\ &= k^2 - 4\\ \end{aligned}$

Subtracting 4 from both sides, we get $x- 4 = (k^2 -4 ) - 4 = k^2 - 8$ .

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Incorrect Choices:

(A)
You will get this wrong answer if you think that $k \times k = k$ , like this: $x = ( k - 2 ) ( k+2 ) = k ^ { 1 } - 4.$ Subtracting 4 from both sides, yields $x - 4 = k - 8$ .

(B)
Tip: Read the entire question carefully.
You will get this wrong answer if you forget the exponent and expand incorrectly, like this: $x = ( k - 2 ) ( k+2 ) = k ^ { 1 } \boxed{+} 4.$ Subtracting 4 from both sides yields $x - 4 = k +4 - 4 = k$ .

(D)
Tip: Read the entire question carefully.
If you solve for $x$ , you will obtain $x = k^2 -4$ . However, the question asks for $x - 4$ instead.

(E)
Tip: $a^2 - b^2 = (a-b)(a+b)$
You will get this wrong answer if you use the wrong formula $a^2 \fbox{+} b^2 = (a-b)(a+b),$ like this:

$x = ( k - 2 ) ( k+2 ) = k ^ { 2 } \boxed{+} 4.$

Subtracting 4 from both sides yields $x - 4 = k^2 +4 - 4 = k^2$ .