Find constants b and c in the polynomial p(x)=x^2+bx+c such that lim x→2 p(x) / x−2=6. Are the constants unique?

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Step 1: Recognize that the limit \( \lim_{x \to 2} \frac{p(x)}{x-2} = 6 \) implies that \( x-2 \) is a factor of \( p(x) \). Therefore, \( p(x) \) can be expressed as \( (x-2)(x-a) \) for some constant \( a \).

Step 2: Expand \( p(x) = (x-2)(x-a) = x^2 - (a+2)x + 2a \).

Step 3: Compare the expanded form \( x^2 - (a+2)x + 2a \) with the original polynomial \( x^2 + bx + c \). This gives the equations: \( b = -(a+2) \) and \( c = 2a \).

Step 4: Use the given limit condition \( \lim_{x \to 2} \frac{p(x)}{x-2} = 6 \). Substitute \( x = 2 \) into the factorized form \( (x-2)(x-a) \) to find \( 2-a = 6 \), which gives \( a = -4 \).

Step 5: Substitute \( a = -4 \) into the equations \( b = -(a+2) \) and \( c = 2a \) to find the values of \( b \) and \( c \). Check if these values are unique by considering the factorization and the limit condition.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this question, we are interested in the limit of the polynomial p(x) as x approaches 2, specifically how it behaves in relation to the expression (x - 2). Understanding limits is crucial for evaluating the continuity and differentiability of functions.

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, p(x) = x^2 + bx + c is a quadratic polynomial. Recognizing the structure of polynomial functions helps in analyzing their limits and determining the values of coefficients that satisfy specific conditions.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. When applying this rule, one differentiates the numerator and denominator until the limit can be resolved. In this problem, if substituting x = 2 results in an indeterminate form, L'Hôpital's Rule may be necessary to find the constants b and c that satisfy the limit condition.