--- title: Polynomial Functions subtitle: Structure, operations, zeros, graphs, identities, and models author: Mathematics Department organization: Your School recipient: Algebra II students date: 2026-09-08 version: "1.1" description: A standards-aligned Algebra II unit on polynomial structure, operations, zeros, graphs, identities, transformations, inequalities, finite differences, and models. keywords: [polynomial functions, Algebra II, rational zero theorem, remainder theorem, factor theorem, complex zeros, binomial theorem, transformations, polynomial inequalities, finite differences] template: assignment paper: us-letter titlePage: false toc: false tocDepth: 2 headingNumbers: false equationNumbers: true columns: 1 figurePlacement: none tableCaptionPosition: top header: Polynomial Functions | Algebra II footer: Student investigation and practice pdfProfile: tagged language: en-US --- # Teacher launch guide > [!IMPORTANT] > **Suggested use:** four 50-minute lessons, or select sections as a modular review packet. Remove this page and the standards note before distributing a student-only copy. ## Learning goals Students will be able to: 1. distinguish polynomials from non-polynomials and describe degree, leading coefficient, and end behavior; 2. use addition, subtraction, multiplication, and division to rewrite polynomial expressions; 3. connect evaluation, remainders, factors, zeros, multiplicity, and graph behavior; 4. find rational and complex zeros and construct polynomials from prescribed zeros; 5. use identities and binomial coefficients without unnecessary expansion; 6. recognize polynomial degree from finite differences and interpret polynomial models; 7. transform polynomial graphs, identify symmetry, and solve polynomial inequalities by sign. ## Standards and mathematical practices This packet targets **CCSS HSA-APR.A.1, HSA-APR.B.2-3, HSA-APR.C.4-5, HSA-APR.D.6, HSN-CN.C.8-9, HSF-BF.B.3**, and **HSF-IF.C.7c**. It emphasizes structure, argument, strategic tool use, and precision. :::table{#tbl:lesson-plan caption="Suggested lesson sequence and evidence of learning"} | Segment | Time | Teacher move | Evidence to collect | | --- | ---: | --- | --- | | Notice and classify | 10 min | Ask what all valid examples have in common | Definition in the student's own words | | Connect representations | 20 min | Link standard, factored, tabular, and graphical forms | Explanation of what each form reveals | | Division and factors | 25 min | Compare long division with evaluation | Correct quotient-remainder identity | | Zeros and graphs | 35 min | Connect multiplicity, end behavior, and turning points | Constrained and justified graph sketch | | Extend the system | 30 min | Introduce rational candidates and conjugate pairs | Complete zero set with verification | | Identities and patterns | 25 min | Compare expansion with structural reasoning | Efficient identity or coefficient argument | | Transform and compare | 25 min | Preserve structure while shifting, scaling, and testing sign | Symmetry or interval argument | | Productive practice | 30 min | Press for reasons, not answer-only work | Annotated solutions and error analysis | | Model and synthesize | 30 min | Separate algebraic results from contextual meaning | Domain restriction and interpretation | | Exit ticket | 10 min | Collect without discussion | Individual mastery evidence | ::: **Prerequisites:** exponent rules, distributive property, factoring quadratics, function notation, and reading intercepts from a graph. **Anticipate these misconceptions:** treating negative exponents as polynomial terms; dropping zero-coefficient placeholders during division; confusing a zero with a y-intercept; and assuming every zero makes a graph cross the axis. # 1. What is a polynomial? A **polynomial in one variable** is a finite sum of terms of the form $a_kx^k$, where every exponent $k$ is a nonnegative integer and every coefficient $a_k$ is a number. In standard form, :::equation{#eq:polynomial-definition alt="P of x equals a sub n x to the n plus a sub n minus one x to the n minus one through a sub zero"} \[ P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\qquad a_n\ne0. \] ::: The degree is $n$; $a_n$ is the leading coefficient; and $a_0$ is the constant term. The zero polynomial is a special case whose degree is usually left undefined. :::table{#tbl:examples-nonexamples caption="Examples make the exponent and finiteness conditions visible"} | Expression | Classification | Reason | | --- | --- | --- | | $4x^5-3x^2+7$ | Polynomial, degree 5 | Finite sum; nonnegative integer exponents | | $-\tfrac12x^3+8x$ | Polynomial, degree 3 | Fractional coefficients are allowed | | $9$ | Polynomial, degree 0 | A nonzero constant is a polynomial | | $x^{-2}+1$ | Not a polynomial in $x$ | Negative exponent | | $\sqrt{x}+3$ | Not a polynomial in $x$ | Exponent $1/2$ | | $\dfrac{2}{x-1}$ | Not a polynomial in $x$ | Variable appears in a denominator | | $1+x+x^2+\cdots$ | Not a polynomial | Infinitely many terms | ::: > [!NOTE] > A coefficient may be a fraction, irrational number, or complex number. It is the exponent restriction, not the appearance of the coefficient, that determines whether the expression is a polynomial in $x$. ## Worked example: read the structure For $P(x)=-2x^4+5x^2-x+9$: - degree: 4; - leading coefficient: $-2$; - constant term: 9; - end behavior: both ends fall because the degree is even and the leading coefficient is negative. ### Check your understanding For each expression, decide whether it is a polynomial in $x$. If it is, state its degree and leading coefficient. If it is not, identify the violated condition. 1. $7-3x+2x^6$ 2. $5x^2-\pi x+1$ 3. $x^3+\dfrac{4}{x}$ 4. $2\sqrt{x}-x^2$ 5. $0$ # 2. Equivalent forms reveal different information The same polynomial function can be written in forms that answer different questions. :::table{#tbl:forms caption="What common polynomial forms make visible"} | Form | Example | What it reveals quickly | | --- | --- | --- | | Standard | $x^3-3x^2+4$ | Degree, leading coefficient, y-intercept | | Factored | $(x+1)(x-2)^2$ | Zeros and multiplicities | | Expanded from a context | $x(12-2x)(9-2x)$ | How quantities in a model are related | | Table or graph | Selected $(x,P(x))$ values | Intercepts, turning behavior, estimates | ::: Because polynomial expressions are closed under addition, subtraction, and multiplication, combining two polynomials with these operations produces another polynomial. ## Worked example: multiply with structure Expand and simplify $(2x-3)(x^2+x+4)$: \[ \begin{aligned} (2x-3)(x^2+x+4) &=2x(x^2+x+4)-3(x^2+x+4)\\ &=2x^3+2x^2+8x-3x^2-3x-12\\ &=2x^3-x^2+5x-12. \end{aligned} \] The product must have degree 3 because the nonzero leading terms $2x$ and $x^2$ multiply to $2x^3$. ## Guided practice 6. Write $3-5x^3+2x-x^4$ in standard form. State its degree and leading coefficient. 7. Simplify $(4x^3-x+6)-(x^3+5x^2-2)$. 8. Expand $(x-4)(x^2+2x-3)$, then check the constant and leading terms before checking the middle terms. 9. Without fully expanding, determine the degree and leading coefficient of $(3x^2-x+1)(-2x^4+7)$. Explain. 10. Find a polynomial $Q(x)$ such that $Q(x)+(2x^2-5x+1)=7x^2+x-4$. # 3. Division, remainders, and factors Polynomial division has the same governing structure as integer division. If $P(x)$ is divided by a nonzero polynomial $D(x)$, then :::equation{#eq:division-algorithm alt="P of x equals D of x times Q of x plus R of x, with the degree of R less than the degree of D"} \[ P(x)=D(x)Q(x)+R(x),\qquad \deg R<\deg D. \] ::: ## Worked example: divide, then interpret Divide $P(x)=x^3-4x^2+x+6$ by $x-2$. \[ \begin{array}{r|rrrr} 2 & 1 & -4 & 1 & 6\\ & & 2 & -4 & -6\\ \hline & 1 & -2 & -3 & 0 \end{array} \] Therefore, \[ x^3-4x^2+x+6=(x-2)(x^2-2x-3)+0=(x-2)(x-3)(x+1). \] The **Remainder Theorem** says that the remainder after division by $x-a$ is $P(a)$. Here $P(2)=0$, so the **Factor Theorem** confirms that $x-2$ is a factor. > [!TIP] > Synthetic division is a compact record of division by $x-a$. Long division remains the general method and also works when the divisor is not linear. ## Guided practice 11. Divide $2x^3+3x^2-11x-6$ by $x+3$. Write the result as divisor times quotient plus remainder. 12. Find the remainder when $P(x)=3x^4-2x^2+5x-7$ is divided by $x-2$ without performing division. 13. Find $k$ so that $x+1$ is a factor of $x^3+kx^2-4x+2$. 14. A student divides $x^3+5x-6$ by $x-1$ using coefficients $1,5,-6$. Explain the error and repair the setup. 15. Is $x-3$ a factor of $2x^4-5x^3-9x^2+18x$? Support the answer with the most efficient calculation. # 4. Zeros, multiplicity, and graph behavior A number $r$ is a **zero** of $P$ exactly when $P(r)=0$. By the Factor Theorem, this happens exactly when $x-r$ is a factor. If $(x-r)^m$ is a factor but $(x-r)^{m+1}$ is not, then $r$ has **multiplicity $m$**. :::figure{#fig:multiplicity caption="The factored function f(x)=(x+1)(x-2)^2 crosses at the simple zero -1 and touches at the double zero 2" alt="Cubic graph crossing the x-axis at negative one and touching it at two" width=90 placement=none} ![Graph of a cubic polynomial with a simple and a double zero](polynomial-zeros.svg) ::: For $f(x)=(x+1)(x-2)^2$: - $x=-1$ has odd multiplicity 1, so the graph crosses the x-axis; - $x=2$ has even multiplicity 2, so the graph touches and turns; - the degree is 3 with positive leading coefficient, so the left end falls and the right end rises. ## A historical boundary worth stating correctly General formulas by radicals exist for degrees 2, 3, and 4. The Abel-Ruffini theorem says there is **no formula by radicals that solves every polynomial equation of degree 5 or greater**. It does not say higher-degree equations are unsolvable: special equations factor exactly, and numerical algorithms approximate roots to high precision. ## Guided practice 16. For $g(x)=-2(x+3)^2(x-1)^3$, list each real zero and multiplicity. Predict whether the graph crosses or touches at each zero and describe both ends. 17. Construct a least-degree polynomial with real coefficients, leading coefficient 2, a double zero at $-1$, and simple zeros at 3 and 5. 18. A fourth-degree polynomial has real zeros $-2$, $0$, and $4$. What can and cannot be concluded about their multiplicities? Give two possible factorizations. 19. Explain why a nonzero polynomial of degree $n$ cannot have more than $n$ distinct zeros. You may use repeated applications of the Factor Theorem. 20. Sketch a possible graph of $h(x)=-(x+2)(x-1)^2(x-4)$. Label intercepts and justify the end behavior. # 5. End behavior and turning points For large $|x|$, the leading term $a_nx^n$ controls a polynomial's behavior. Lower-degree terms become comparatively small. :::table{#tbl:end-behavior caption="End behavior from degree parity and leading-coefficient sign"} | Degree | Leading coefficient | As $x\to-\infty$ | As $x\to\infty$ | | --- | --- | --- | --- | | Even | Positive | $P(x)\to\infty$ | $P(x)\to\infty$ | | Even | Negative | $P(x)\to-\infty$ | $P(x)\to-\infty$ | | Odd | Positive | $P(x)\to-\infty$ | $P(x)\to\infty$ | | Odd | Negative | $P(x)\to\infty$ | $P(x)\to-\infty$ | ::: A degree-$n$ polynomial has at most $n$ real zeros and at most $n-1$ turning points. These are upper bounds, not promises. ## Worked example: constrain a sketch before plotting For $q(x)=-3x^5+2x^2-7$: - degree 5 and negative leading coefficient imply left end up, right end down; - $q(0)=-7$, so the y-intercept is $(0,-7)$; - the graph has at most five x-intercepts and at most four turning points; - these facts constrain a sketch but do not locate the zeros. ## Quick reasoning 21. Describe the end behavior of $4x^6-9x+2$ and $-2x^7+5x^3$. 22. Can a degree-4 polynomial have exactly five turning points? Exactly three? Explain. 23. Give two polynomials with the same end behavior but different numbers of real zeros. 24. A graph has left end down and right end up. What must be true about the degree parity and leading-coefficient sign? # 6. Finding rational and complex zeros For an integer-coefficient polynomial, the **Rational Zero Theorem** gives candidates. If $p/q$ in lowest terms is a rational zero of $a_nx^n+\cdots+a_0$, then $p$ divides $a_0$ and $q$ divides $a_n$. A candidate is not a zero until substitution or division verifies it. ## Worked example: search, verify, finish Solve $2x^3-3x^2-8x+12=0$. The candidates are \[ \pm1,\ \pm2,\ \pm3,\ \pm4,\ \pm6,\ \pm12, \quad \pm\frac12,\ \pm\frac32. \] Since $P(2)=0$, \[ 2x^3-3x^2-8x+12=(x-2)(2x^2+x-6)=(x-2)(2x-3)(x+2). \] The zeros are $2$, $3/2$, and $-2$. The **Fundamental Theorem of Algebra** says a degree-$n$ polynomial has exactly $n$ complex zeros when multiplicity is counted. For real coefficients, nonreal zeros occur in conjugate pairs. ## Worked example: construct from zeros The least-degree monic polynomial with zeros $3$ and $1+2i$ and real coefficients must also have zero $1-2i$: \[ \begin{aligned} P(x)&=(x-3)(x-(1+2i))(x-(1-2i))\\ &=(x-3)((x-1)^2+4)\\ &=x^3-5x^2+11x-15. \end{aligned} \] ## Guided practice 25. List all rational-zero candidates for $3x^3+2x^2-12x-8$. 26. Find all zeros of $x^3-4x^2-7x+10$, given that one zero is 1. 27. Find all zeros of $x^4-5x^2+4$ without using the quartic formula. 28. A real cubic has zeros $-2$ and $3+i$. Find its third zero and write a monic polynomial in standard form. 29. Construct a least-degree real polynomial with a double zero at 4 and zeros $-1+2i$ and $-1-2i$. 30. Explain why a real polynomial of odd degree must have at least one real zero. 31. Explain why a Rational Zero Theorem candidate list may contain no actual zero. # 7. Polynomial identities and binomial structure An **identity** is true for every allowed value of the variable. An equation such as $x^2=9$ is true only for selected values. \[ a^2-b^2=(a-b)(a+b), \qquad a^3-b^3=(a-b)(a^2+ab+b^2). \] The Binomial Theorem organizes every power $(a+b)^n$: :::equation{#eq:binomial-theorem alt="The sum from k equals zero to n of n choose k times a to the n minus k times b to the k"} \[ (a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k. \] ::: ## Worked example: one coefficient, no full expansion In $(2x-3)^5$, the $x^3$ term occurs when $k=2$: \[ \binom52(2x)^3(-3)^2=720x^3. \] ## Guided practice 32. Verify $(x+y)^3=x^3+3x^2y+3xy^2+y^3$ by multiplication. 33. Factor $8x^3-27$ completely over the integers. 34. Find the coefficient of $x^4$ in $(x-2)^6$ without fully expanding. 35. Is $(x+1)^2=x^2+1$ an identity? Diagnose the missing term. 36. Prove that $P(x)-P(1)$ is divisible by $x-1$ for every polynomial $P$. # 8. Investigation: one polynomial, four representations Consider \[ P(x)=x^3-4x^2+x+6. \] Do not begin by graphing. Build the representations in an order that lets each result check the next. 1. Calculate $P(-1)$, $P(0)$, $P(1)$, $P(2)$, and $P(3)$. **[5 points]** 2. Use one result to identify a linear factor. **[2 points]** 3. Divide to obtain a quadratic factor, then factor completely. **[5 points]** 4. State each zero and its multiplicity. **[3 points]** 5. Predict the graph's behavior at each zero and its left and right end behavior. **[4 points]** 6. Sketch the graph. Your sketch must agree with all five evaluated points. **[4 points]** 7. Which representation made the repeated zero easiest to detect: the table, standard form, factored form, or graph? Defend your choice. **[3 points]** :::keep > [!NOTE] > A graphing tool may be used only after question 6. If the graph disagrees with your reasoning, locate and annotate the error rather than replacing your work. ::: :::figure{#fig:reasoning-cycle caption="A verification cycle for polynomial reasoning" alt="A cycle from conjecture to algebra, graph, and revision" width=78 placement=none} ```mermaid flowchart LR A[Conjecture from structure] --> B[Verify algebraically] B --> C[Compare with a graph] C --> D{Representations agree?} D -- Yes --> E[Explain the connection] D -- No --> F[Locate the mismatch] F --> B ``` ::: ::::regions{tracks="1 1" gutter=18} :::region # 9. Productive practice Show reasoning. An unsupported answer may be correct without demonstrating the intended learning goal. ## Core 37. Classify $4x^7-\sqrt{3}x^2+1$. Then classify $4x^7-\sqrt{x}+1$. 38. Simplify: \[ (2x^3-x^2+4)-(5x^3+3x-7). \] 39. Multiply $(x+2)(2x^2-5x+3)$. 40. Divide $x^3+2x^2-5x-6$ by $x+3$. 41. Use the Remainder Theorem to find the remainder when $2x^5-x^3+4x-9$ is divided by $x+1$. ::: :::region ## Reason and connect 42. Find all zeros of $x^3-2x^2-5x+6$, given that $x=1$ is a zero. 43. Write a polynomial in standard form with zeros $-2$, $1$, and $1$, and y-intercept $-8$. 44. Decide whether this claim is always, sometimes, or never true: "A degree-4 polynomial has four x-intercepts." Justify. 45. Two polynomials of degree at most 3 agree at four distinct x-values. Explain why they must be the same polynomial. 46. Create a polynomial whose graph crosses at $x=-3$, touches at $x=1$, and rises on both ends. State its least possible degree. ## Error analysis Maya writes: \[ \frac{x^3-8}{x-2}=x^2+4 \] 47. Without doing full division, show that her quotient cannot be correct. 48. Repair the quotient and verify your result by multiplication. ::: :::: # 10. Modeling challenge: an open-top box A $16$-inch by $10$-inch sheet of card is used to make an open-top box. Squares of side length $x$ are cut from each corner, and the sides are folded up. ## Build the model 49. Explain why the box dimensions are $x$, $16-2x$, and $10-2x$. 50. Write the volume function in factored form and expand it into standard form. 51. State the physically meaningful domain. Explain why the algebraic domain of the polynomial is larger. :::equation{#eq:box-volume alt="V of x equals x times sixteen minus two x times ten minus two x"} \[ V(x)=x(16-2x)(10-2x). \] ::: ## Analyze 52. Find $V(1)$, $V(2)$, $V(3)$, and $V(4)$. Which cut gives the largest volume among these choices? 53. Explain the meaning of the zeros $0$, $5$, and $8$ in the formula. Which belong to the boundary of the physical domain? 54. A numerical tool estimates a maximum near $x=2.0$. Check that this is plausible using values at $x=1.9$, $2.0$, and $2.1$, and report the resulting volume to the nearest cubic inch. 55. Why would giving only the algebraic zeros be an incomplete answer to a question about the box? > [!TIP] > Mathematical modeling includes interpreting restrictions. A formula can accept inputs that the physical situation cannot. # 11. Finite differences: detecting polynomial degree For equally spaced x-values, a degree-$n$ polynomial has constant $n$th finite differences. Constant first differences indicate a linear pattern; constant second differences indicate a quadratic pattern; constant third differences indicate a cubic pattern. :::table{#tbl:finite-differences caption="Values of a quadratic and their finite differences"} | $x$ | $P(x)$ | First difference | Second difference | | ---: | ---: | ---: | ---: | | 0 | 1 | | | | 1 | 4 | 3 | | | 2 | 9 | 5 | 2 | | 3 | 16 | 7 | 2 | | 4 | 25 | 9 | 2 | ::: Here $P(x)=x^2+2x+1$. The constant second difference is $2=2!$ times the leading coefficient. ## Guided practice 56. The values $2,7,16,29,46$ occur at $x=0,1,2,3,4$. Build a difference table and identify the likely degree. 57. Find a quadratic matching the values in question 56. 58. Explain why constant second differences do not prove a pattern remains quadratic outside the observed data unless a polynomial model is assumed. 59. A polynomial sequence has constant third difference 12. What is its leading coefficient? 60. Create five consecutive values from a cubic polynomial and exchange them with a partner. The partner must recover the degree from differences. # 12. Transformations and symmetry For $Q(x)=aP(b(x-h))+k$: - $h$ shifts the graph horizontally; $k$ shifts it vertically; - $a$ scales vertically and reflects across the x-axis when $a<0$; - $b$ scales horizontally by $1/|b|$ and reflects across the y-axis when $b<0$. An **even** polynomial satisfies $P(-x)=P(x)$ and has y-axis symmetry. It contains only even powers. An **odd** polynomial satisfies $P(-x)=-P(x)$ and has origin symmetry. It contains only odd powers and has constant term 0. A polynomial may be neither. ## Quick reasoning 61. Describe every transformation from $y=P(x)$ to $y=-2P(x-3)+1$. 62. Classify each as even, odd, or neither: $4x^6-3x^2+8$ and $x^5-2x^3+x$. 63. Explain why $x^4+x$ is neither even nor odd. 64. If $P$ has degree 4 and leading coefficient 3, state the degree and leading coefficient of $P(x+2)-5$. 65. If the zeros of $P$ are $r_1,r_2,r_3$, state the zeros of $P(x-4)$. # 13. Polynomial inequalities by sign A polynomial's sign is constant between consecutive real zeros. At a zero of odd multiplicity, the sign changes. At a zero of even multiplicity, it does not. **Worked example.** Solve $(x+2)(x-1)^2(x-4)>0$. The critical numbers are $-2,1,4$. The sign changes at $-2$ and $4$, but not at the double zero $1$. Testing one interval gives \[ (x+2)(x-1)^2(x-4)>0\quad\text{on}\quad(-\infty,-2)\cup(4,\infty). \] ## Guided practice 66. Solve $(x-3)(x+1)<0$. 67. Solve $(x+2)^2(x-5)\le0$. 68. Explain algebraically why an even-multiplicity zero does not change the sign. 69. A positive-leading quartic has simple zeros $-3$ and $2$ and a double zero $0$. Where is it negative? 70. Construct a monic polynomial with zeros $-2$ and $3$ that is nonnegative for every real $x$. # Exit ticket Complete independently in the final 8 minutes. 1. Is $3x^4-2x+\sqrt{7}$ a polynomial? State its degree. 2. Find the remainder when $x^3-5x+1$ is divided by $x-2$. 3. If $P(4)=0$, state one factor of $P(x)$. 4. Describe the graph behavior at a zero of multiplicity 2. 5. In one sentence, explain the logical connection among a zero, a factor, and an x-intercept. 6. List the rational-zero candidates for $2x^3-3x+6$. 7. Find the coefficient of $x^3$ in $(x+1)^5$. 8. The values $3,8,15,24$ occur at consecutive integer inputs. What do the finite differences suggest about degree? 9. Describe the transformation from $P(x)$ to $P(x+4)-2$. 10. Solve $(x-1)(x+5)>0$. --- ## Student reflection Circle one: **ready to apply / need one more example / need a conference** The idea I can explain most clearly is: ______________________________________ The question I still have is: ______________________________________________