Question

If one zero of the polynomial \(x^2 - 5x - c\) is (-1), find the value of c. Also, find the other zero.

Hint:

For polynomial \(x^2 + bx + c\), the sum of zeroes is \(-b\) and the product is \(c\). This shortcut only works when the coefficient of \(x^2\) is 1.

Answer 1

Step 1: Using the Zero Condition:
Since (-1) is a zero of the polynomial \(x^2 - 5x - c\), substitute \(x = -1\) into the polynomial.

Step 2: Substitute and Simplify:
\[ (-1)^2 - 5(-1) - c = 0 \] \[ 1 + 5 - c = 0 \] \[ 6 - c = 0 \] \[ c = 6 \]
Step 3: Form the New Polynomial:
Substitute \(c = 6\) into the polynomial:
\[ x^2 - 5x - 6 \]
Step 4: Factorize to Find the Other Zero:
We need two numbers whose product is \(-6\) and sum is \(-5\).
Those numbers are \(-6\) and \(+1\).

\[ x^2 - 5x - 6 = (x - 6)(x + 1) \]
Step 5: Find the Zeros:
\[ x - 6 = 0 \Rightarrow x = 6 \] \[ x + 1 = 0 \Rightarrow x = -1 \]
Final Answer:
The value of \(c\) is 6.
The other zero of the polynomial is 6.