Problem Statement
To determine what must be subtracted from the polynomial \( P(x) = x^4 + 2x^3 - 13x^2 - 12x + 218 \) such that the resulting polynomial is divisible by \( x^2 - 4x + 3 \).
Step 1: Factor the Divisor
The polynomial \( x^2 - 4x + 3 \) can be factored as:
- (x - 1)(x - 3)
This means that \( P(x) \) must equal zero at \( x = 1 \) and \( x = 3 \) for divisibility.
Step 2: Calculate P(1) and P(3)
Next, we will evaluate \( P(1) \) and \( P(3) \):
- For \( x = 1 \):
- P(1) = 1^4 + 2(1^3) - 13(1^2) - 12(1) + 218
- = 1 + 2 - 13 - 12 + 218
- = 196
- For \( x = 3 \):
- P(3) = 3^4 + 2(3^3) - 13(3^2) - 12(3) + 218
- = 81 + 54 - 117 - 36 + 218
- = 200
Step 3: Find the Value to Subtract
To make \( P(x) \) divisible by \( x^2 - 4x + 3 \), we need to ensure that both evaluations equal zero.
- To adjust for \( P(1) \):
- Subtract 196 from \( P(x) \)
- To adjust for \( P(3) \):
- Subtract 200 from \( P(x) \)
Since we want a single adjustment, we can find the average of these two values:
- Average = (196 + 200) / 2 = 198
Thus, we should subtract 198 from \( P(x) \).
Conclusion
To ensure \( P(x) \) is divisible by \( x^2 - 4x + 3 \), subtract:
- 198 from the polynomial \( x^4 + 2x^3 - 13x^2 - 12x + 218 \).

(2x + 194)should be subtracted