ID: 371cbf6b
(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Advanced Math)
Comment: Focus on the term in question, not on solving the whole thing.
Method 1: Because the question asks for the value of ab, it is crucial to understand how the "a" and "b" terms will come together after distributing.
ax * -bx = -abx^2
It can be tempting to pair this product with -9x^2 on the right-hand side of the given equation, but 9 is surprisingly not an answer. (Incorrect options often spring from intermediate numbers you might encounter while working toward a solution.) The question you should be asking is What other terms will combine to yield x^2? Well, that has to be 3 and 5x^2.
3 * 5x^2 = 15x^2
These will be the only x^2 terms that come from the distribution. The following must therefore be true:
15x^2 - abx^2 = -9x^2
Since x^2 is attached to all terms, we can safely divide it out and work with one fewer variable.
15 - ab = -9
-ab = -24
The correct answer must be (C).
Method 2: Generally, Desmos is less useful when you are dealing with more unknowns than the typical "x" and "y." Sure, you can add a slider for an extra unknown, but when you start to get into more than one slider, you will probably have to either work out part of the problem on your own or resort to a lot of inefficient guessing and checking. I imagine that for most students, deriving the value of "a" would not prove too challenging.
ax * 5x^2 = 20x^3
5ax^3 = 20x^3
5a = 20
a = 4
The expression now appears more manageable.
((4)x + 3)(5x^2 - bx + 4) = 20x^3 - 9x^2 - 2x + 12
As shown above, in method 1, you could now focus on the x^2 term exclusively.
-4bx^2 + 15x^2 = -9x^2
-4b + 15 = -9
-4b = -24
b = 6
If a = 4 and b = 6, then ab = 24, and the correct answer is (C). An alternative but similar approach would be to plug in the expression (4x + 3)(5x^2 - bx + 4) and set it equal to "y," creating a second equation with the right-hand side of the given equation. When the graphs overlapped, you would know you had the correct value of "b."
Method 3: Although you would be doing a lot of extra work for no reason, if you wanted to practice your distribution, you could multiply everything out and then answer the question. I would recommend creating a 2 * 3 (or 3 * 2) grid and filling in each product in its corresponding square, as you would do in a Punnett square in biology.
Notice that by using this method, like terms will be easy to connect diagonally; I have marked such relationships with a dotted line in the diagram. Now, you can combine like terms—I will group using parentheses where doing so might add clarity—and work out what is what.
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