Using the remainder theorem, find the remainder when x^4 - 3x^3 + 2x^2 - x - 1 is divided by (x - 2).
Understand the Problem
The question is asking to find the remainder when the polynomial x^4 - 3x^3 + 2x^2 - x - 1 is divided by (x - 2) using the remainder theorem, which requires evaluating the polynomial at the root of the divisor.
Answer
The remainder is \( -3 \).
Answer for screen readers
The remainder when ( x^4 - 3x^3 + 2x^2 - x - 1 ) is divided by ( x - 2 ) is ( -3 ).
Steps to Solve
-
Identify the polynomial and divisor
We are given the polynomial ( p(x) = x^4 - 3x^3 + 2x^2 - x - 1 ) and need to find the remainder when it is divided by ( x - 2 ).
-
Apply the Remainder Theorem
According to the Remainder Theorem, the remainder of a polynomial ( p(x) ) when divided by ( x - a ) is equal to ( p(a) ).
Here, ( a = 2 ).
-
Evaluate ( p(2) )
Substitute ( x = 2 ) in the polynomial: [ p(2) = (2)^4 - 3(2)^3 + 2(2)^2 - (2) - 1 ]
Simplifying step by step:
- ( (2)^4 = 16 )
- ( 3(2)^3 = 3 \times 8 = 24 )
- ( 2(2)^2 = 2 \times 4 = 8 )
Combine: [ p(2) = 16 - 24 + 8 - 2 - 1 ]
-
Calculate the value of ( p(2) )
Continuing with the evaluation: [ p(2) = 16 - 24 + 8 - 2 - 1 = 16 - 24 = -8 \ -8 + 8 = 0 \ 0 - 2 = -2 \ -2 - 1 = -3 ]
Thus, ( p(2) = -3 ).
-
Determine the remainder
Therefore, the remainder when ( p(x) ) is divided by ( x - 2 ) is ( -3 ).
The remainder when ( x^4 - 3x^3 + 2x^2 - x - 1 ) is divided by ( x - 2 ) is ( -3 ).
More Information
The Remainder Theorem simplifies the process of finding remainders in polynomial division, allowing for quick calculations by simply evaluating the polynomial at the root of the divisor.
Tips
- Failing to correctly substitute the value into the polynomial.
- Miscalculating powers, especially with negative signs.
- Forgetting to combine the terms step by step, leading to miscalculated totals.
AI-generated content may contain errors. Please verify critical information
