Could you further explain your explanation of Practice Question #4 in PWN the SAT Math Guide “Polynomials” chapter (p. 148). In your long division confirmation of the answer (explanation p. 326), how do you know to divide by 9x + 1?

Sure. Here’s the question:

First, as I say in the solution, all you really need to do here is pay attention to the leading terms. When you divide 18x+(\text{something}) by ax+(\text{something}) and get 2x+(\text{something}), then a must be 9.

When I check by doing long division, I use 9x+1 because I’m trying to confirm that the equation works if a=9. As shown in the solution, the long division of \dfrac{18x^2-16x+3}{9x+1} works out to 2x-2 with a remainder of 5—or 2x-2+\dfrac{5}{9x+1}. That matches the right hand side of the original equation.

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