Question

type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar. if p - 1 is a factor of p⁴ + p² + p - k, the value of k is .

Explanation:

Step1: Recall Factor Theorem

If \( p - a \) is a factor of a polynomial \( f(p) \), then \( f(a)=0 \). Here, \( a = 1 \) (since the factor is \( p - 1 \)), so we substitute \( p = 1 \) into the polynomial \( f(p)=p^{4}+p^{2}+p - k \) and set it equal to 0.

Step2: Substitute \( p = 1 \) into the polynomial

Substitute \( p = 1 \) into \( f(p) \):
\( f(1)=(1)^{4}+(1)^{2}+(1)-k \)
Simplify the powers: \( 1^{4}=1 \), \( 1^{2}=1 \), so \( f(1)=1 + 1+1 - k \)
Combine like terms: \( f(1)=3 - k \)

Step3: Solve for \( k \)

Since \( p - 1 \) is a factor, \( f(1)=0 \). So:
\( 3 - k = 0 \)
Add \( k \) to both sides: \( 3=k \)

Answer:

3