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Difficulty: EasyPolynomials, Factor and Remainder Theorems

What is the remainder when the polynomial P(x)=x3−4x2+5x−2P(x) = x^3 - 4x^2 + 5x - 2 is divided by (x+1)(x + 1)?

  1. −12-12Answer
  2. B
    00
  3. C
    −4-4
  4. D
    −2-2

Answer

−12-12
According to the Remainder Theorem, dividing a polynomial P(x)P(x) by a linear divisor (x−a)(x - a) yields a remainder equal to P(a)P(a). Here, the divisor is (x+1)(x + 1), which corresponds to a=−1a = -1. Substituting x=−1x = -1 into P(x)=x3−4x2+5x−2P(x) = x^3 - 4x^2 + 5x - 2 gives (−1)3−4(−1)2+5(−1)−2=−1−4−5−2=−12(-1)^3 - 4(-1)^2 + 5(-1) - 2 = -1 - 4 - 5 - 2 = -12. Thus, the value −12-12 is correct.

Step-by-Step Solution

1
Apply the Remainder Theorem
To find the remainder when P(x)P(x) is divided by (x+1)(x + 1), evaluate P(−1)P(-1) by setting x+1=0  ⟹  x=−1x + 1 = 0 \implies x = -1.
By the Remainder Theorem, dividing P(x)P(x) by (x−a)(x - a) leaves a remainder equal to P(a)P(a).
2
Substitute x=−1x = -1 into the polynomial P(x)=x3−4x2+5x−2P(x) = x^3 - 4x^2 + 5x - 2
P(−1)=(−1)3−4(−1)2+5(−1)−2P(-1) = (-1)^3 - 4(-1)^2 + 5(-1) - 2
Replace each instance of xx with −1-1.
3
Simplify the powers and terms
P(−1)=−1−4(1)−5−2=−1−4−5−2=−12P(-1) = -1 - 4(1) - 5 - 2 = -1 - 4 - 5 - 2 = -12
Compute arithmetic operations following standard order of operations.

Key Concept

Remainder Theorem
Estimated Time:45s
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Topics

Polynomials, Factor and Remainder Theorems

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