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Difficulty: MediumEquivalent Algebraic Expressions

If the expression 3x2−bx+10x−2\frac{3x^2 - bx + 10}{x - 2} is equivalent to 3x−c−12x−23x - c - \frac{12}{x - 2} for all x≠2x \neq 2, where bb and cc are positive constants, what is the value of bb?

Answer: 17

Answer

17
Multiplying both sides of the equivalence by x−2x - 2 results in 3x2−bx+10=(3x−c)(x−2)−123x^2 - bx + 10 = (3x - c)(x - 2) - 12. Expanding the right side gives 3x2−(6+c)x+(2c−12)3x^2 - (6 + c)x + (2c - 12). By equating the coefficients of corresponding terms, we get 2c−12=102c - 12 = 10, which solves to c=11c = 11. Substituting this into the xx-coefficient equivalence −b=−(6+c)-b = -(6 + c) gives b=6+11=17b = 6 + 11 = 17.

Step-by-Step Solution

1
Multiply both sides of the expression by the denominator x−2x - 2 to equate the numerators.
3x2−bx+10=(3x−c)(x−2)−123x^2 - bx + 10 = (3x - c)(x - 2) - 12
Since the rational expressions are equivalent for all x≠2x \neq 2, their numerators must be equal when written over a common denominator.
2
Expand and simplify the right side of the equation.
3x2−bx+10=3x2−(6+c)x+(2c−12)3x^2 - bx + 10 = 3x^2 - (6 + c)x + (2c - 12)
Expanding (3x−c)(x−2)(3x - c)(x - 2) yields 3x2−6x−cx+2c3x^2 - 6x - cx + 2c, and subtracting 1212 gives the simplified polynomial expression.
3
Equate the constant terms to solve for cc.
10=2c−12⇒c=1110 = 2c - 12 \Rightarrow c = 11
For the polynomials to be equivalent, their constant terms must be equal.
4
Equate the coefficients of the xx terms to solve for bb.
b=6+c⇒b=6+11=17b = 6 + c \Rightarrow b = 6 + 11 = 17
Equating the coefficients of xx gives −b=−(6+c)-b = -(6 + c), which simplifies to b=6+cb = 6 + c.

Key Concept

Equivalence of rational and polynomial expressions via coefficient comparison

Alternative Method

Alternatively, you can evaluate the equivalence at a convenient value of xx. For instance, substituting x=0x = 0 into the expression gives −5=−c+6-5 = -c + 6, yielding c=11c = 11. Then, evaluating the equation at another convenient value such as x=1x = 1 allows you to solve for bb directly using the now-known value of cc.
Estimated Time:1m 30s
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Topics

Equivalent Algebraic Expressions

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