Wage Bill Remainder D3353B

1. **State the problem:** We are given the remainders when $W(x)$ is divided by $(x-500)$ and $(x+400)$, and we want to find the remainder when $W(x)$ is divided by $x^2 - 100x - 200000$. Then, based on this remainder at the current basic fee $x=20000$, decide if the wage increase to $25000$ should be approved. 2. **Recall the Remainder Theorem:** - The remainder when a polynomial $W(x)$ is divided by $(x-a)$ is $W(a)$. - Since $x^2 - 100x - 200000 = (x-500)(x+400)$, the remainder when dividing by this quadratic is a linear polynomial $R(x) = Ax + B$. 3. **Use given remainders to find $R(x)$:** - Given $W(500) = 160000$ (remainder when divided by $x-500$) - Given $W(-400) = 70000$ (remainder when divided by $x+400$) Since $R(x)$ is the remainder when dividing by $(x-500)(x+400)$, it must satisfy: $$R(500) = W(500) = 160000$$ $$R(-400) = W(-400) = 70000$$ Let $R(x) = Ax + B$. Then: $$A(500) + B = 160000$$ $$A(-400) + B = 70000$$ 4. **Solve the system for $A$ and $B$:** Subtract second equation from first: $$500A + B - (-400A + B) = 160000 - 70000$$ $$500A + B - (-400A) - B = 90000$$ $$500A + 400A = 90000$$ $$900A = 90000$$ $$A = \frac{90000}{900} = 100$$ Substitute $A=100$ into first equation: $$100 \times 500 + B = 160000$$ $$50000 + B = 160000$$ $$B = 160000 - 50000 = 110000$$ So, $$R(x) = 100x + 110000$$ 5. **Evaluate remainder at current basic fee $x=20000$:** $$R(20000) = 100 \times 20000 + 110000 = 2000000 + 110000 = 2110000$$ 6. **Compare remainder with threshold $2650000$:** Since $2110000 < 2650000$, the remainder is less than the threshold. 7. **Conclusion:** Based on the current basic fee, the remainder condition is satisfied, so the committee can approve the wage increase from $20000$ to $25000$.