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Question the-eq

2.4 Equivalent Expressions - Simplifying polynomials
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open parenthesis, a x plus 3, close parenthesis, times, open parenthesis, 5 x squared, minus b x, plus 4, close parenthesis, equals, 20 x cubed, minus 9 x squared, minus 2 x, plus 12

The equation above is true for all x, where a and b are constants. What is the value of ab ?

A.

18

B.

20

C.

24

D.

40

The equation above is true for all x , where a and b are constants. What

Hard-difficulty · SAT Math · Equivalent Expressions — Simplifying polynomials. Read the question above, select your answer, and check the full explanation below to understand exactly why the correct choice works.

Answer explanation

Choice C is correct. If the equation is true for all x, then the expressions on both sides of the equation will be equivalent. Multiplying the polynomials on the left-hand side of the equation gives 5 a, x cubed, minus a, b x squared, plus 4 a, x, plus 15 x squared, minus 3 b x, plus 12. On the right-hand side of the equation, the only x squared-term is negative 9 x squared. Since the expressions on both sides of the equation are equivalent, it follows that negative a, b x squared, plus 15 x squared, equals negative 9 x squared, which can be rewritten as open parenthesis, negative a, b plus 15, close parenthesis, times x squared, equals negative 9 x squared. Therefore, negative a, b plus 15, equals negative 9, which gives a, b equals 24.

Choice A is incorrect. If a, b equals 18, then the coefficient of x squared on the left-hand side of the equation would be negative 18 plus 15, equals negative 3, which doesn’t equal the coefficient of x squared, negative 9, on the right-hand side. Choice B is incorrect. If a, b equals 20, then the coefficient of x squared on the left-hand side of the equation would be negative 20 plus 15, equals negative 5, which doesn’t equal the coefficient of x squared, negative 9, on the right-hand side. Choice D is incorrect. If a, b equals 40, then the coefficient of x squared on the left-hand side of the equation would be negative 40 plus 15, equals negative 25, which doesn’t equal the coefficient of x squared, negative 9, on the right-hand side.

 

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