Making a Given Number a Zero of a Polynomial

Video Explanation

Question

What should be subtracted from the polynomial

\[ f(x) = x^2 – 16x + 30 \]

so that \(15\) is a zero of the resulting polynomial?

Options:

(a) 30    (b) 14    (c) 15    (d) 16

Solution

Step 1: Use the Zero Condition

If \(15\) is a zero of the new polynomial, then its value at \(x = 15\) must be zero.

Step 2: Find the Value of the Given Polynomial at \(x = 15\)

\[ f(15) = 15^2 – 16(15) + 30 \]

\[ = 225 – 240 + 30 \]

\[ = 15 \]

Step 3: Find What Must Be Subtracted

To make the value zero, we must subtract \(15\).

\[ 15 – 15 = 0 \]

Conclusion

The number that must be subtracted is:

\[ \boxed{15} \]

Hence, the correct option is (c) 15.