Making a Given Number a Zero of a Polynomial

Video Explanation

Question

What should be added to the polynomial

\[ f(x) = x^2 – 5x + 4 \]

so that \(3\) is a zero of the resulting polynomial?

Options:

(a) 1    (b) 2    (c) 4    (d) 5

Solution

Step 1: Use the Zero Condition

If \(3\) is a zero of the new polynomial, then its value at \(x = 3\) must be zero.

Step 2: Find the Value of the Given Polynomial at \(x = 3\)

\[ f(3) = 3^2 – 5(3) + 4 \]

\[ = 9 – 15 + 4 \]

\[ = -2 \]

Step 3: Find What Must Be Added

To make the value zero, we must add \(2\).

\[ -2 + 2 = 0 \]

Conclusion

The number that must be added is:

\[ \boxed{2} \]

Hence, the correct option is (b) 2.