Hello! Do you think you could explain number 8 in the Polynomials Practice Questions of the PWN the SAT Math Guide 4th Edition? Thank you!

Gladly! I think the best way to go for this one is to factor.

    \begin{align*}f(x)&=3x^3+9x^2-30x\\f(x)&=3x(x^2+3x-10)\\f(x)&=3x(x+5)(x-2)\end{align*}

    \begin{align*}g(x)&=x^2+3x-10\\g(x)&=(x+5)(x-2)\end{align*}

From that, you can see pretty quickly that two answer choices aren’t right. First, you see that g(x) is most definitely a factor of f(x), so A isn’t the answer. You can also see that g(x) has two real zeros (–5 and 2), so C isn’t the answer.

You can also probably see what the answer is. Once everything is factored, it’s clear that (x-5) is not a factor of f(x). That means B is the answer.

As for D, well, once you know the answer is B you might not want to worry about it. But in case it’s keeping you up at night, here’s the deal. If h(x)=f(x)+2g(x), then we can use the factoring we’ve already done to write the following.

    \begin{align*}h(x)&=3x(x+5)(x-2)+2(x+5)(x-2)\end{align*}

See how you can factor (x+5)(x-2) out of both terms?

    \begin{align*}h(x)&=3x[(x+5)(x-2)]+2[(x+5)(x-2)]\\h(x)&=(3x+2)[(x+5)(x-2)]\end{align*}

That’s how you know that (3x+2) is a factor of h(x), which means you can cross off choice D.

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