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Difficulty: MediumPolynomial Factors and Graphs

A polynomial f(x)f(x) has a remainder of 1212 when divided by x−4x - 4. If g(x)=(x−2)f(x−2)−5g(x) = (x - 2)f(x - 2) - 5, what is the remainder when g(x)g(x) is divided by x−6x - 6?

  1. A
    7
  2. B
    19
  3. 43Answer
  4. D
    -53

Answer

The correct answer is 43, which is the remainder when g(x) is divided by x - 6.
According to the Remainder Theorem, since the polynomial f(x)f(x) has a remainder of 1212 when divided by x−4x - 4, we have f(4)=12f(4) = 12. To find the remainder when g(x)g(x) is divided by x−6x - 6, we apply the Remainder Theorem again to evaluate g(6)g(6). Substituting x=6x = 6 into the definition of g(x)g(x) gives g(6)=(6−2)f(6−2)−5=4f(4)−5g(6) = (6 - 2)f(6 - 2) - 5 = 4f(4) - 5. Substituting f(4)=12f(4) = 12 yields 4(12)−5=48−5=434(12) - 5 = 48 - 5 = 43.

Step-by-Step Solution

1
Apply the Remainder Theorem to the polynomial f(x)f(x).
f(4)=12f(4) = 12
The Remainder Theorem states that the remainder when a polynomial f(x)f(x) is divided by x−cx - c is equal to f(c)f(c).
2
Set up the expression for the remainder of g(x)g(x) when divided by x−6x - 6.
Evaluate g(6)g(6)
By the Remainder Theorem, the remainder of g(x)g(x) divided by x−6x - 6 is equal to g(6)g(6).
3
Substitute x=6x = 6 into the definition of g(x)=(x−2)f(x−2)−5g(x) = (x - 2)f(x - 2) - 5.
g(6)=(6−2)f(6−2)−5=4f(4)−5g(6) = (6 - 2)f(6 - 2) - 5 = 4f(4) - 5
Evaluating g(6)g(6) requires substituting x=6x = 6 into all occurrences of xx in the equation for g(x)g(x).
4
Substitute the value of f(4)f(4) into the expression for g(6)g(6) and simplify.
g(6)=4(12)−5=48−5=43g(6) = 4(12) - 5 = 48 - 5 = 43
Using the value of f(4)=12f(4) = 12 from Step 1 allows us to calculate the numerical value of the remainder.

Key Concept

The Remainder Theorem and evaluation of composite polynomial functions.

Alternative Method

Instead of evaluating g(6)g(6) directly, one can write f(x)=(x−4)q(x)+12f(x) = (x - 4)q(x) + 12 for some polynomial q(x)q(x). Substituting this expression into the equation for g(x)g(x) gives g(x)=(x−2)[(x−6)q(x−2)+12]−5=(x−2)(x−6)q(x−2)+12(x−2)−5g(x) = (x - 2)[(x - 6)q(x - 2) + 12] - 5 = (x - 2)(x - 6)q(x - 2) + 12(x - 2) - 5. Evaluating this expression at x=6x = 6 makes the first term 00, leaving 12(6−2)−5=12(4)−5=4312(6 - 2) - 5 = 12(4) - 5 = 43.
Estimated Time:1m 30s
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Topics

Polynomial Factors and Graphs

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