If x - 4 is a factor of x2 - x - k, then k is

Analysis

To solve this problem, we use the Factor Theorem. The Factor Theorem states that if $(x - a)$ is a factor of a polynomial $P(x)$, then $P(a) = 0$.

Concept

In this question, we are given:

• The polynomial $P(x) = x^2 - x - k$
• The factor $(x - 4)$

According to the Factor Theorem, if $(x - 4)$ is a factor, then substituting $x = 4$ into the polynomial must result in zero ($P(4) = 0$).

Step-by-Step Solution

1. Set up the equation: Since $(x - 4)$ is a factor, we set $x - 4 = 0$, which gives $x = 4$.

2. Substitute $x = 4$ into the polynomial: $P(4) = (4)^2 - (4) - k = 0$

3. Simplify the expression: $16 - 4 - k = 0$ $12 - k = 0$

4. Solve for $k$: $12 = k$ Therefore, $k = 12$.

Conclusion

By substituting the root of the factor into the quadratic expression, we found that $k$ must be 12 for the expression to be divisible by $(x - 4)$.

Exam Tip: Whenever you see "is a factor of," immediately think of the Factor Theorem. Just set the factor to zero to find the value of $x$, plug it into the equation, and solve for the unknown variable. It saves you the stress of long division!