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Question
\[ (x^2 + bx  2)(x + 3) = x^3 + 6x^2 + 7x 6 \]
In the equation above, b is a constant. If the equation is true for all values of x, what is the value of b?
A) 2 B) 3 C) 7 D) 9 Answer
To find the value of b, expand the lefthand side of the equation and then collect like terms so that the lefthand side is in the same form as the righthand side.
\[(x^2 + bx − 2)(x + 3) = (x^3 + bx^2 − 2x) + (3x^2 + 3bx − 6) \]
\[ = x3 + (3 + b)x^2 + (3b − 2)x − 6 \]
Since the two polynomials are equal for all values of x, the coefficient of matching powers of x should be the same. Therefore, and x^3 + 6x^2 + 7x − 6 reveals that 3 + b = 6 and 3b − 2 = 7. Solving either of these equations gives b = 3, which is choice B.
Questions may also ask you to use structure to rewrite expressions. The expression may be of a particular type, such as a difference of squares, or it may require insightful analysis.
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