Examkin

Question

Difficulty: MediumPolynomials, Factor and Remainder Theorems

Given that (x−3)(x - 3) is a factor of the polynomial P(x)=x3+kx2+mx−6P(x) = x^3 + kx^2 + mx - 6 and that dividing P(x)P(x) by (x+1)(x + 1) leaves a remainder of −12-12, what is the value of k+mk + m?

  1. −1-1Answer
  2. B
    −7-7
  3. C
    −5-5
  4. D
    55

Answer

The value of k+mk + m is −1-1.
Applying the Factor Theorem P(3)=0P(3) = 0 gives 3k+m=−73k + m = -7, and applying the Remainder Theorem P(−1)=−12P(-1) = -12 gives k−m=−5k - m = -5. Solving these simultaneous linear equations gives k=−3k = -3 and m=2m = 2, which sums to k+m=−1k + m = -1.

Step-by-Step Solution

1
Apply the Factor Theorem for the linear factor (x−3)(x - 3)
3k+m=−73k + m = -7
By the Factor Theorem, if (x−3)(x - 3) is a factor, then P(3)=0P(3) = 0. Substituting x=3x = 3 gives 33+k(3)2+m(3)−6=0  ⟹  27+9k+3m−6=0  ⟹  9k+3m=−21  ⟹  3k+m=−73^3 + k(3)^2 + m(3) - 6 = 0 \implies 27 + 9k + 3m - 6 = 0 \implies 9k + 3m = -21 \implies 3k + m = -7.
2
Apply the Remainder Theorem for the divisor (x+1)(x + 1)
k−m=−5k - m = -5
By the Remainder Theorem, dividing P(x)P(x) by (x+1)(x + 1) gives remainder P(−1)=−12P(-1) = -12. Substituting x=−1x = -1 gives (−1)3+k(−1)2+m(−1)−6=−12  ⟹  −1+k−m−6=−12  ⟹  k−m−7=−12  ⟹  k−m=−5(-1)^3 + k(-1)^2 + m(-1) - 6 = -12 \implies -1 + k - m - 6 = -12 \implies k - m - 7 = -12 \implies k - m = -5.
3
Solve the system of linear equations for kk and mm
k=−3k = -3 and m=2m = 2
Adding the two equations (3k+m)+(k−m)=−7+(−5)(3k + m) + (k - m) = -7 + (-5) gives 4k=−12  ⟹  k=−34k = -12 \implies k = -3. Substituting k=−3k = -3 into k−m=−5k - m = -5 gives −3−m=−5  ⟹  m=2-3 - m = -5 \implies m = 2.
4
Calculate the target value k+mk + m
k+m=−1k + m = -1
Summing k=−3k = -3 and m=2m = 2 yields k+m=−3+2=−1k + m = -3 + 2 = -1.

Key Concept

Factor and Remainder Theorems
Estimated Time:1m 30s
Rate this question

Topics

Polynomials, Factor and Remainder Theorems

More questions from this topic