Question

1. given (f(x)=x^{2}+kx - 19), and the remainder when (f(x)) is divided by (x - 7) is - 5, then what is the value of (k?)

Explanation:

Step1: Apply Remainder Theorem

According to the Remainder Theorem, if \(f(x)=x^{2}+kx - 19\) is divided by \(x - 7\), then \(f(7)\) is the remainder.
\[f(7)=7^{2}+7k-19\]

Step2: Set up the equation

Since the remainder is - 5, we set \(f(7)=-5\). So, \(7^{2}+7k - 19=-5\).
\[49 + 7k-19=-5\]
\[30 + 7k=-5\]

Step3: Solve for k

Subtract 30 from both sides:
\[7k=-5 - 30\]
\[7k=-35\]
Divide both sides by 7:
\[k=\frac{-35}{7}=-5\]

Answer:

\(k = - 5\)