PART I · SHORT RESPONSE第一部分 · 短答题SAT-style MCQ + ON/BC short answer · 22 marksSAT 风格选择题 + 安/卑省考短答 · 共 22 分
Section A · Short ResponseA 部分 · 短答题
Mix of multiple-choice and short-answer items. For MCQs, circle the letter; show enough work in the margin that a marker could verify. For division items, line the columns up carefully and label your quotient and remainder explicitly. No calculator on Q1–Q5.本节包含选择题与短答题。选择题请圈出字母答案，并在空白处写下足以让阅卷人核对的过程。除法题需将列对齐整齐，并明确标出商与余数。Q1–Q5 不可使用计算器。
Let $p(x) = x^{3} - 4x^{2} + 5x - 7$. By the remainder theorem, what is the remainder when $p(x)$ is divided by $(x - 2)$?设 $p(x) = x^{3} - 4x^{2} + 5x - 7$。根据余数定理，$p(x)$ 被 $(x - 2)$ 除时的余数是多少？
(A) $-5$
(B) $-3$
(C) $0$
(D) $5$
Q3MEDIUM中🇺🇸 US美SAT-style MCQSAT 风格选择题§5 End Behaviour末端行为 · HSF-IF.B.4[4 marks][4 分]
As $x \to +\infty$ and as $x \to -\infty$, what is the end behaviour of $f(x) = -2x^{4} + 6x^{3} - x + 8$?当 $x \to +\infty$ 与 $x \to -\infty$ 时，$f(x) = -2x^{4} + 6x^{3} - x + 8$ 的末端行为是什么？
(A)$f(x) \to +\infty$ on both ends.两端都有 $f(x) \to +\infty$。
(B)$f(x) \to -\infty$ on both ends.两端都有 $f(x) \to -\infty$。
(C)$f(x) \to +\infty$ as $x \to -\infty$ and $f(x) \to -\infty$ as $x \to +\infty$.$x \to -\infty$ 时 $f(x) \to +\infty$；$x \to +\infty$ 时 $f(x) \to -\infty$。
(D)$f(x) \to -\infty$ as $x \to -\infty$ and $f(x) \to +\infty$ as $x \to +\infty$.$x \to -\infty$ 时 $f(x) \to -\infty$；$x \to +\infty$ 时 $f(x) \to +\infty$。
(a)Use synthetic division to divide $p(x)$ by $(x - 1)$. Show the synthetic-division tableau with the divisor root, the bring-down row, and the multiply/add columns.用综合除法将 $p(x)$ 除以 $(x - 1)$。列出综合除法竖式，包括除式根、下移行和乘 / 加列。[3]
(b)State the quotient $q(x)$ and the remainder $r$.写出商 $q(x)$ 与余数 $r$。[2]
(c)Use your remainder from (b) to state whether $(x - 1)$ is a factor of $p(x)$, citing the factor theorem by name.利用 (b) 中的余数说明 $(x - 1)$ 是否为 $p(x)$ 的因式，并指明所用为因式定理。[1]
PART II · EXTENDED RESPONSE第二部分 · 简答题AP-feeder FRQ + honors · 36 marksAP 衔接简答题 + 荣誉级 · 共 36 分
Section B · Extended ResponseB 部分 · 简答题
Show every algebraic step. State the theorem you invoke by name: "by the factor theorem…", "by the rational root theorem the candidates are…". When sketching, label every $x$-intercept with its multiplicity and state the end-behaviour signs as a one-line conclusion. No calculator on Q6–Q9.每一步代数运算都要写出。引用所用定理时务必写明名称："由因式定理……"、"由有理根定理，候选根为……"。作图时为每个 $x$ 轴截距标注其重数，并用一句话给出末端行为符号。Q6–Q9 不可使用计算器。
(a)Evaluate $p(-1)$, $p(2)$, and $p(-3)$.求 $p(-1)$、$p(2)$ 和 $p(-3)$ 的值。[3]
(b)Citing the factor theorem by name, state which of $(x + 1)$, $(x - 2)$, and $(x + 3)$ are factors of $p(x)$.引用因式定理，指出 $(x + 1)$、$(x - 2)$、$(x + 3)$ 中哪几个是 $p(x)$ 的因式。[2]
(c)Use your results from (a) and (b) to factor $p(x)$ completely over the real numbers, and state all three real zeros of $p$.利用 (a) 和 (b) 的结果，在实数范围内将 $p(x)$ 完全因式分解，并写出 $p$ 的三个实零点。[3]
(a)List the complete set of candidate rational roots given by the rational root theorem. Show how you generated the $\pm p / q$ list from the constant term and the leading coefficient.列出由有理根定理给出的所有候选有理根。说明如何从常数项与首项系数生成 $\pm p / q$ 候选清单。[3]
(b)Test candidates by direct substitution (or synthetic division) until you find one rational root. Identify it.通过直接代入（或综合除法）逐个检验候选根，直至找到一个有理根，写出该根。[2]
(c)Use synthetic or long division to depress the cubic to a quadratic factor, and then factor the quadratic.用综合除法或长除法将三次式降次为一个二次因式，再对该二次式进行因式分解。[3]
(d)State all three real roots of the equation and the complete factorization of $2x^{3} - 3x^{2} - 11x + 6$ over the rationals.写出该方程的三个实根，并给出 $2x^{3} - 3x^{2} - 11x + 6$ 在有理数范围内的完全因式分解。[2]
(a)State the degree of $f$ and the sign of its leading coefficient. Hence state the end behaviour as $x \to \pm \infty$.写出 $f$ 的次数以及首项系数的符号，进而写出当 $x \to \pm \infty$ 时的末端行为。[2]
(b)List every $x$-intercept of $y = f(x)$ together with its multiplicity, and state for each whether the graph crosses or touches the $x$-axis there.列出 $y = f(x)$ 的所有 $x$ 轴截距及其重数，并指出在该点处图象是穿过还是相切于 $x$ 轴。[3]
(c)State the $y$-intercept of the graph.写出图象的 $y$ 轴截距。[1]
(d)Sketch a smooth curve consistent with parts (a)–(c). Label every intercept and the end-behaviour arrows.作出与 (a)–(c) 一致的光滑曲线，标出每个截距及末端行为箭头。[3]
A real cubic polynomial $p(x)$ has integer coefficients, leading coefficient $1$, and is known to have $x = 3$ and $x = 1 + 2i$ as two of its three roots.实系数三次多项式 $p(x)$ 的系数均为整数，首项系数为 $1$，已知其三个根中有 $x = 3$ 和 $x = 1 + 2i$ 两个。
(a)Using the complex conjugate root theorem (cite HSN-CN.A.7 for US, or BC PC12 polynomial-equations for BC), state the third root and justify in one sentence why it must take that value.利用复共轭根定理（美国引用 HSN-CN.A.7，卑诗引用 PC12 多项式方程），写出第三个根，并用一句话说明为何必为该值。[2]
(b)Show that the product $(x - (1 + 2i))(x - (1 - 2i))$ simplifies to the real quadratic factor $x^{2} - 2x + 5$.证明乘积 $(x - (1 + 2i))(x - (1 - 2i))$ 化简后等于实二次因式 $x^{2} - 2x + 5$。[3]
(c)Hence write $p(x)$ as a product of a real linear and a real irreducible quadratic factor, and expand it into standard form $p(x) = x^{3} + a x^{2} + b x + c$ with integer coefficients $a, b, c$.据此将 $p(x)$ 写成一个实一次因式与一个实不可约二次因式的乘积，并展开成标准形式 $p(x) = x^{3} + a x^{2} + b x + c$，其中 $a, b, c$ 为整数。[3]
(d)Interpret the result geometrically: how many real $x$-intercepts does the graph of $y = p(x)$ have, and what does the irreducible quadratic factor tell you about the graph?从几何上解释该结果：$y = p(x)$ 的图象有多少个实 $x$ 轴截距？不可约二次因式又告诉你图象的哪个特征？[1]
PART III · MODELING / APPLIED第三部分 · 建模 / 应用Universal · 32 marks通用 · 共 32 分
Section C · Modeling and ApplicationsC 部分 · 建模与应用
Name your variables (with units) before writing equations. Set up your model, solve, and conclude with a one-sentence answer in context. Calculator permitted throughout Part III.在写方程之前先命名变量（并标明单位）。先建立模型、再求解，最后用一句话结合情境作答。第三部分全程可用计算器。
An open-top box is built from a $20\text{ cm} \times 16\text{ cm}$ rectangle of card by cutting a square of side $x$ cm from each of the four corners and folding up the flaps. The volume of the resulting box, in cm$^{3}$, is given by $V(x) = x(20 - 2x)(16 - 2x)$.将一张 $20\text{ cm} \times 16\text{ cm}$ 的长方形卡纸的四个角各剪掉一个边长为 $x$ cm 的正方形，再把翼片折起，做成一个无盖纸盒。所得纸盒的体积（cm$^{3}$）为 $V(x) = x(20 - 2x)(16 - 2x)$。
(a)Expand $V(x)$ into standard polynomial form and state its degree.将 $V(x)$ 展开为标准多项式形式，并写出其次数。[2]
(b)State the practical (situational) domain of $V(x)$ in interval notation, and explain in one sentence why each endpoint is excluded.用区间记号写出 $V(x)$ 的实际（情境）定义域，并用一句话说明为何两个端点都被排除。[2]
(c)Solve $V(x) = 384$ algebraically using the rational root theorem to find any rational root, then factoring. Report every solution in the practical domain, rounded to the nearest tenth if irrational.用代数方法求解 $V(x) = 384$：先用有理根定理找到一个有理根，再因式分解。报告所有落在实际定义域内的解，若为无理数则保留至小数点后一位。[4]
(d)Pick one solution from (c) and state, in one sentence, the dimensions ($L \times W \times H$, in cm) of the resulting box.从 (c) 的解中任选一个，用一句话写出所得纸盒的尺寸（长 $\times$ 宽 $\times$ 高，单位 cm）。[2]
Q11MEDIUM中🇨🇦 ON安ON Provincial-style安大略省考风格§5 Building a Polynomial from its Zeros由零点构造多项式 · MHF4U Polynomial & Rational FunctionsMHF4U 多项式与有理函数[11 marks][11 分]
A cubic polynomial function $f$ models a measured signal. The function has $x$-intercepts at $x = -2$ (a simple zero), $x = 1$ (a zero of multiplicity 2), and a $y$-intercept of $f(0) = 8$.用一个三次多项式函数 $f$ 拟合一段测量信号。该函数的 $x$ 轴截距为 $x = -2$（单根）与 $x = 1$（重数为 2 的零点），$y$ 轴截距为 $f(0) = 8$。
(a)Write $f(x)$ in factored form $f(x) = a(x - r_{1})(x - r_{2})^{m_{2}}$ with unknown leading coefficient $a$.把 $f(x)$ 写成因式形式 $f(x) = a(x - r_{1})(x - r_{2})^{m_{2}}$，其中首项系数 $a$ 待定。[2]
(b)Use the condition $f(0) = 8$ to determine the value of $a$.利用条件 $f(0) = 8$ 确定 $a$ 的值。[2]
(c)Expand $f(x)$ into standard form $f(x) = a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0}$.将 $f(x)$ 展开为标准形式 $f(x) = a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0}$。[3]
(d)State the end behaviour of $f$ as $x \to \pm \infty$, the multiplicity-based crossing/touching behaviour at each zero, and sketch a smooth curve consistent with all of the above. Label intercepts and end-behaviour arrows.写出 $f$ 当 $x \to \pm \infty$ 时的末端行为，以及每个零点处由重数决定的"穿过 / 相切"行为，并作出与上述全部信息一致的光滑曲线。标出截距与末端行为箭头。[4]
Consider the polynomial inequality $x^{3} - 4x^{2} + x + 6 \le 0$.考虑多项式不等式 $x^{3} - 4x^{2} + x + 6 \le 0$。
(a)Verify by direct substitution that $x = -1$ is a root of $p(x) = x^{3} - 4x^{2} + x + 6$, citing the factor theorem in one sentence.通过直接代入验证 $x = -1$ 是 $p(x) = x^{3} - 4x^{2} + x + 6$ 的根，并用一句话引用因式定理。[2]
(b)Use synthetic or long division to factor $p(x)$ completely over the real numbers. State all three real zeros.用综合除法或长除法在实数范围内将 $p(x)$ 完全因式分解，并写出三个实零点。[3]
(c)Construct a sign chart for $p(x)$ on the intervals determined by its zeros. Label each interval with the sign of $p$.在由零点划分的各区间上构建 $p(x)$ 的符号表，标出每个区间上 $p$ 的符号。[3]
(d)State the solution set of $x^{3} - 4x^{2} + x + 6 \le 0$ in interval notation, taking care to include the boundary zeros.用区间记号写出 $x^{3} - 4x^{2} + x + 6 \le 0$ 的解集，注意把边界零点包含在内。[2]
(e)A small reservoir's net inflow rate, in cubic metres per hour, $t$ hours after midnight is modeled by $r(t) = -p(t) = -(t^{3} - 4t^{2} + t + 6)$ for $t \in [0, 4]$. Use your sign analysis from (c) to state the subinterval(s) of $[0, 4]$ on which water is flowing into the reservoir (i.e. $r(t) \ge 0$).某小型水库的净流入速率（立方米每小时）在零点后 $t$ 小时由 $r(t) = -p(t) = -(t^{3} - 4t^{2} + t + 6)$ 给出，其中 $t \in [0, 4]$。利用 (c) 中的符号分析，写出 $[0, 4]$ 中水流入水库的子区间（即 $r(t) \ge 0$ 的部分）。[1]
🇺🇸 US Common Core美国共同核心HSA-SSE.A.1 · HSA-SSE.A.2 · HSA-CED.A.1 · HSA-REI.D.11 · HSF-IF.B.4 · HSF-IF.C.7 · HSF-IF.C.8 · HSF-LE.A.3 · HSF-BF.B.3