AMC 10A 2025 (Problem 7)Suppose aaa and bbb are real numbers. When the polynomial x3+x2+ax+bx^3+x^2+ax+bx3+x2+ax+b is divided by x−1x-1x−1, the remainder is 444. When the polynomial is divided by x−2x-2x−2, the remainder is 666. What is b−ab-ab−a?(A) 14\text{(A)}\;14(A)14(B) 15\text{(B)}\;15(B)15(C) 16\text{(C)}\;16(C)16(D) 17\text{(D)}\;17(D)17(E) 18\text{(E)}\;18(E)18Related TopicsCoreToolkit 32 — Remainder of Polynomial DivisionHints (3)Hint 1By Toolkit 32 — Remainder of Polynomial Division:x−1=0⟹x=1x-1=0\Longrightarrow x=1x−1=0⟹x=113+12+a+b=41^3+1^2+a+b=413+12+a+b=4a+b=2a+b=2a+b=2Hint 2By Toolkit 32 — Remainder of Polynomial Division:x−2=0⟹x=2x-2=0\Longrightarrow x=2x−2=0⟹x=223+22+2a+b=62^3+2^2+2a+b=623+22+2a+b=62a+b=−62a+b=-62a+b=−6Hint 3From the two equations:{a+b=22a+b=−6⟹a=−8, b=10 \begin{cases} a+b=2\\ 2a+b=-6 \end{cases} \Longrightarrow a=-8,\ b=10 {a+b=22a+b=−6⟹a=−8, b=10Therefore,b−a=18b-a=18b−a=18Final Answer(E) 181818Related Problems (3)AMC 10B 2022 (Problem 21)AMC 12A 2022 (Problem 21)AMC 12B 2021 Spring (Problem 20)