Finding the Value of k Using Zero of a Polynomial

Video Explanation

Question

If one of the zeroes of the quadratic polynomial

\[ f(x) = (k – 1)x^2 + kx + 1 \]

is \(-3\), find the value of \(k\).

Solution

Step 1: Use the Zero Condition

If \(-3\) is a zero of the polynomial, then

\[ f(-3) = 0 \]

Step 2: Substitute \(x = -3\)

\[ (k – 1)(-3)^2 + k(-3) + 1 = 0 \]

\[ 9(k – 1) – 3k + 1 = 0 \]

Step 3: Simplify

\[ 9k – 9 – 3k + 1 = 0 \]

\[ 6k – 8 = 0 \]

\[ 6k = 8 \Rightarrow k = \frac{4}{3} \]

Conclusion

The value of \(k\) is:

\[ \boxed{\frac{4}{3}} \]