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\[ (x^2 + bx - 2)(x + 3) = x^3 + 6x^2 + 7x -6 \]
In the equation above, b is a constant. If the equation is true for all values of x, what is the value of b?
To find the value of b, expand the left-hand side of the equation and then collect like terms so that the left-hand side is in the same form as the right-hand 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.