If [tex]$p-1$[/tex] is a factor of [tex]$p^4+p^2+p-k$[/tex], what is the value of [tex]$k$[/tex]?
Answer (2)
Since p − 1 is a factor of p 4 + p 2 + p − k , substitute p = 1 into the polynomial.
Evaluate the polynomial at p = 1 : 1 4 + 1 2 + 1 − k = 0 .
Simplify the equation: 3 − k = 0 .
Solve for k : k = 3 $.
Explanation
Understanding the Factor Theorem We are given that p − 1 is a factor of the polynomial p 4 + p 2 + p − k . This means that if we substitute p = 1 into the polynomial, the result must be zero. This is because if p − 1 is a factor, then p 4 + p 2 + p − k = ( p − 1 ) × q ( p ) for some polynomial q ( p ) . When p = 1 , we have ( 1 − 1 ) \tims q ( 1 ) = 0 .
Substituting p=1 Now, substitute p = 1 into the polynomial p 4 + p 2 + p − k and set it equal to zero:
( 1 ) 4 + ( 1 ) 2 + ( 1 ) − k = 0
Simplifying the Equation Simplify the equation:
1 + 1 + 1 − k = 0
3 − k = 0
Solving for k Solve for k :
k = 3
Final Answer Therefore, the value of k is 3.
Examples
Understanding polynomial factors is crucial in many engineering applications. For instance, when designing filters for signal processing, engineers use polynomials to describe the filter's behavior. Knowing the factors of these polynomials helps them predict and control the filter's response to different frequencies. Similarly, in control systems, understanding polynomial factors aids in analyzing system stability and designing controllers that ensure smooth and predictable operation. These applications highlight how polynomial factorization is not just an abstract mathematical concept but a practical tool for solving real-world engineering problems.
The value of k is 3 because substituting p = 1 into the polynomial p 4 + p 2 + p − k must equal zero. This follows from the Factor Theorem, which states that if p − 1 is a factor, then the polynomial evaluates to zero at that point. Therefore, 3 − k = 0 leads us to k = 3 .
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