$P(x)=x^4-2 x^3+k x-4$, where $k$ is an unknown integer. $P(x)$ divided by $(x-1)$ has a remainder of 0. What is the value of $k$?
Answer (2)
Substitute x = 1 into the polynomial P ( x ) since ( x − 1 ) is a factor.
Simplify the equation P ( 1 ) = 1 4 − 2 ( 1 ) 3 + k ( 1 ) − 4 = 0 .
Solve for k : 1 − 2 + k − 4 = 0 ⇒ k = 5 .
The value of k is 5 .
Explanation
Understanding the Problem We are given the polynomial P ( x ) = x 4 − 2 x 3 + k x − 4 , where k is an unknown integer. We know that when P ( x ) is divided by ( x − 1 ) , the remainder is 0. This means that ( x − 1 ) is a factor of P ( x ) , and therefore, P ( 1 ) = 0 . Our goal is to find the value of k .
Substituting x=1 into P(x) Since P ( 1 ) = 0 , we can substitute x = 1 into the expression for P ( x ) and set it equal to 0: P ( 1 ) = ( 1 ) 4 − 2 ( 1 ) 3 + k ( 1 ) − 4 = 0
Solving for k Now we simplify the equation and solve for k :
1 − 2 + k − 4 = 0 k − 5 = 0 k = 5
Final Answer Therefore, the value of k is 5.
Examples
Polynomials are used to model curves and relationships in various fields. For example, engineers use polynomials to design bridges and other structures. If a bridge's height is modeled by a polynomial P ( x ) , and we know that the bridge touches the ground at x = 1 , then P ( 1 ) = 0 . Finding the unknown coefficients of the polynomial, like k in our problem, helps ensure the bridge's stability and safety. This problem demonstrates a basic application of the factor theorem, which is crucial in polynomial analysis and real-world applications.
The value of k in the polynomial P ( x ) = x 4 − 2 x 3 + k x − 4 , which allows ( x − 1 ) to be a factor, is 5. This is determined by setting P ( 1 ) = 0 . Therefore, k = 5 .
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