Algebraic functions and polynomials: synthesis and challenges

Summary

This lesson closes the unit on algebraic functions and polynomials. It gathers into a single deductive chain what was proved in its nine lessons, from the concept of function and the division algorithm to the complete analysis of rational functions and of functions with radicals; it shows how the unit rests on the natural numbers and on the number systems of the earlier units, states honestly the results that were admitted without proof and explains how it connects with the analytic geometry of the plane. It ends with ten challenges, harder than the proposed problems of the unit, which require combining several lessons and finding an idea.

Learning objectives

Reconstruct the chain of definitions and theorems of the unit, explaining which result rests on which.
Relate the theorems on polynomials to their analogues on integers studied in the earlier units.
Distinguish the results proved in the unit from those that were admitted and will be proved in other courses.
Solve problems that combine several lessons of the unit and require a non-mechanical idea, among them the formulation of a concrete situation by means of polynomials.

What the unit achieved

Functions and their operations

The unit began with the concept of function: a rule that assigns to each element of the domain exactly one image. In the lesson on algebraic functions the rules that determine the domain of a formula were fixed, the lemma on the sign of a product was proved, the basis of the whole later study of signs, and functions were classified as polynomial, rational, irrational and transcendental; that the classification, which refers to functions and not to their formulas, is not empty was shown by the proof that $\frac{1}{x}$ coincides with no polynomial, and the lesson on the algebra of polynomials later proved that $\sqrt{x}$ is not a rational function. The lesson on function operations treated real functions as objects. From the lesson on sets, relations and functions of the first unit it inherited the associativity of composition and the theorem according to which a function has an inverse if and only if it is bijective; it added what is specific to real functions: the domain of a composition, with which composition remains associative but not commutative; that a strictly monotonic function is bijective onto its range and has a monotonic inverse, which makes it possible to invert by restriction the functions that are not injective; that the graph of the inverse is the reflection in the line $y = x$ ; and that every function with a symmetric domain is, in a unique way, the sum of an even function and an odd one.

Polynomials and their factorization

The core of the unit is the theory of polynomials, which is developed as a chain of theorems, each one resting on the previous one.

The division algorithm, proved in the lesson on the algebra of polynomials, gives a unique quotient and a unique remainder. With divisor $x - r$ , the remainder is $P(r)$ (remainder theorem), and from this follows the factor theorem: $r$ is a root if and only if $x - r$ divides $P$ .
Applied to several roots, the factor theorem gives the lemma on distinct roots: a nonzero polynomial of degree $n$ has at most $n$ roots. From it the identity principle is deduced: two polynomials define the same function if and only if they have the same coefficients. Since up to that moment polynomials had been treated through their coefficients, the order is not circular; and from then on each polynomial is identified with its function, coefficients can be compared, two polynomials that coincide at infinitely many points are equal, and it is proved that the square root is not a rational function.
To find the roots, the lesson on factoring quadratic polynomials completed the square and obtained the quadratic formula, the discriminant and Vieta’s formulas; the lesson on the roots of higher-degree polynomials added synthetic division, the rational root theorem, multiplicity and the bounds on the roots.
Once the fundamental theorem of algebra was admitted, and with the conjugate roots theorem, the factorization over the reals was proved: every real polynomial is its leading coefficient times factors $x - r$ and times irreducible quadratic factors.

P(x) = a_n (x - r_1) \cdots (x - r_k) \left( x^2 + p_1 x + s_1 \right) \cdots \left( x^2 + p_m x + s_m \right), \qquad k + 2m = n

From the quotient to the graph

With the factorization in hand, the second half of the unit studied quotients. The lesson on algebraic fractions showed that cancelling a common factor is an equality only on the original domain, characterized the equality of two fractions and proved the existence and uniqueness of the partial fraction decomposition when the denominator has only linear factors: with distinct factors, the coefficients are obtained by evaluating at the roots an identity which, by the identity principle, holds also where the fractions are not defined; with repeated factors, uniqueness requires isolating at each root the term of highest power. The lesson on sign charts and inequalities extended the rule of signs to $n$ factors and read off from the factorization the sign of a polynomial or of a quotient: the sign changes when crossing a root of odd multiplicity, is preserved at one of even multiplicity, and the irreducible factors do not count. From this followed the solution of inequalities and of domains and the distinction between vertical asymptotes and removable points.

The last two lessons brought all the tools together. In the lesson on rational functions, division writes $\frac{P}{Q}$ as a polynomial $C$ plus a proper fraction that tends to zero, and $C$ is the horizontal, oblique or curvilinear asymptote; the sign chart decides in which direction the branches escape, and the discriminant of the equation $y \cdot Q(x) = P(x)$ gives the range when that equation is of degree $2$ . In the lesson on functions with radicals, the equality $\sqrt{x^2} = \abs{x}$ forces one to treat separately the large positive and the large negative values of the variable, and one and the same function can have two different asymptotes.

What was admitted

The unit has been explicit about what it did not prove. The fundamental theorem of algebra was admitted, whose proofs require tools from analysis. The arrows of limits, with which asymptotes and the behavior of a function near a point or at infinity are described, were used with an intuitive meaning: what was proved was the sign of the function on each side and the decomposition that separates the asymptote, and what was described was the approach itself. It was admitted that rational functions and functions with radicals are continuous on each interval of their domain, and, in its general form, the intermediate value theorem: a function continuous on an interval $\icc{a}{b}$ takes every value between $f(a)$ and $f(b)$ , and in particular vanishes if $f(a)$ and $f(b)$ have opposite signs, which is Bolzano’s theorem; with it, zeros that algebra cannot reach were located and ranges were computed. The existence and uniqueness of the partial fraction decomposition when there are irreducible quadratic factors were admitted, since they require Bézout’s identity for polynomials, as was the uniqueness of the factorization over the reals, on which it depends that the least common multiple of several polynomials is determined up to a constant factor. Finally, it was admitted that every function defined by an algebraic expression satisfies a polynomial equation in $x$ and $y$ , a fact on which no other result depends. And the maxima and minima that are not revealed by an algebraic transformation remained open. Limits, continuity, the intermediate value theorem, extrema and the integration of partial fractions are studied in the courses Calculus I and Calculus II; the fundamental theorem of algebra, by contrast, is proved in a course on complex analysis.

The following figure gathers the three segments into a single chain. Each node of the column is a definition or a theorem, numbered according to the logical order and joined by an arc to the earlier links that its proof uses; around it float the supports, that is, the principles and tools to which the chain resorts again and again, each one joined by a solid line to the links that rest on it. All of them are admitted or come from the earlier units, and that is why they are drawn as rings: the results that the unit proves and reuses, such as the lemma on the sign of a product or the factor theorem, appear as links, joined by arcs to those that use them. Clicking a support highlights all the links that use it, and each link refers to the lesson in which it is proved.

Supports of the chain

AInduction. A property that holds in a base case and passes from each case to the next, or from all the previous ones to the next, always holds; proved in the unit on the natural numbers. Admitted without proof in this unit.
BEuclid’s lemma. If a prime divides a product of integers, it divides one of the factors; it follows that, if an integer divides a product and is coprime to one of the factors, it divides the other; proved in the unit on the natural numbers. Admitted without proof in this unit.
CComposition and inverse. Composition of functions is associative and preserves injectivity, and a function has an inverse if and only if it is bijective, and that inverse is unique; proved in the lesson on sets, relations and functions. Admitted without proof in this unit.
DConjugate roots. If a complex number is a root of a polynomial with real coefficients, so is its conjugate; proved in the unit on the number systems. Admitted without proof in this unit.
EFundamental theorem of algebra. Every polynomial of degree $n \geq 1$ with complex coefficients has at least one complex root; admitted without proof. Admitted without proof in this unit.
FBézout and unique factorization. If two polynomials have no common factors of positive degree, there exist polynomials $U$ and $V$ with $U A + V B = 1$ ; and the factorization of a real polynomial into linear factors and irreducible quadratic factors is unique up to order; both are admitted, because the course does not develop the greatest common divisor of polynomials. Admitted without proof in this unit.
GLimits and continuity. The arrows $x \to r$ and $f(x) \to \infty$ have an intuitive meaning, and rational functions and functions with radicals are continuous on each interval of their domain; admitted until the course Calculus I. Admitted without proof in this unit.
HIntermediate value. If a function is continuous on an interval $\icc{a}{b}$ , it takes on it every value between $f(a)$ and $f(b)$ ; in particular, if $f(a)$ and $f(b)$ have opposite signs, it vanishes at some point of $\ioo{a}{b}$ (Bolzano’s theorem); admitted until the course Calculus I. Admitted without proof in this unit.

Segment 1 Functions and their operations
- Define the function and fix the rules that determine the domain of a formula.
- Prove the lemma on the sign of a product, the basis of the later study of signs.
- Classify algebraic functions and prove that the classification is not empty.
- Treat real functions as objects: domain of the composition, inverse by restriction to a set on which the function is monotonic, graph of the inverse and decomposition into even and odd parts.
1. Function and domain. A function assigns to each element of its domain exactly one image; the domain of a formula excludes the zeros of the denominators and the negative radicands of the roots of even index.
  Proved in: Algebraic functions of a real variable.
2. Sign of a product. For real numbers $a$ and $b$ , $ab = 0$ if and only if $a = 0$ or $b = 0$ , and $ab > 0$ if and only if both are nonzero and of the same sign; if $b \neq 0$ , the quotient $\frac{a}{b}$ has the sign of $ab$ .
  Proved in: Algebraic functions of a real variable.
3. A quotient that is not a polynomial. No polynomial function coincides with $\frac{1}{x}$ at all nonzero real numbers; hence there are rational functions that are not polynomial.
  Uses: 1. Function and domain. Proved in: Algebraic functions of a real variable.
4. Graphs of powers and roots. For $n \geq 2$ , the functions $x^n$ and $\sqrt[n]{x}$ preserve the order of the nonnegative real numbers, $x^n < x < \sqrt[n]{x}$ if $0 < x < 1$ , and the power is even or odd according as $n$ is.
  Uses: 1. Function and domain. Proved in: Algebraic functions of a real variable.
5. Domain of the composition. For real functions, the domain of $f \circ g$ consists of the $x$ in the domain of $g$ whose image lies in the domain of $f$ ; with these domains composition is associative, and it is not commutative.
  Uses: 1. Function and domain. Rests on: C (Composition and inverse). Proved in: Function operations, inverse functions and graph transformations.
6. Monotonic functions. Every strictly increasing or strictly decreasing function is injective, and the sum of two increasing functions is increasing.
  Uses: 1. Function and domain. Proved in: Function operations, inverse functions and graph transformations.
7. Inverse of a monotonic function. A function strictly increasing on a set $A$ of real numbers is bijective from $A$ onto its range, and its inverse is strictly increasing; the same holds for strictly decreasing ones. In this way the functions that are not injective are inverted by restricting the domain.
  Uses: 6. Monotonic functions. Rests on: C (Composition and inverse). Proved in: Function operations, inverse functions and graph transformations.
8. Graph of the inverse. The point $(a, b)$ lies on the graph of $f$ if and only if $(b, a)$ lies on that of $f^{-1}$ ; hence the graph of the inverse is the reflection of that of $f$ in the line $y = x$ .
  Uses: 7. Inverse of a monotonic function. Rests on: C (Composition and inverse). Proved in: Function operations, inverse functions and graph transformations.
9. Even and odd parts. The product of two even functions or of two odd ones is even, and that of an even one and an odd one is odd; every function with a symmetric domain is, in a unique way, the sum of an even function and an odd one.
  Uses: 1. Function and domain. Proved in: Function operations, inverse functions and graph transformations.
Segment 2 Polynomials and their factorization
- Prove the division algorithm and deduce from it the remainder and factor theorems.
- Bound the number of roots of a polynomial and obtain the identity principle.
- Find the roots: quadratic formula, discriminant, Vieta’s formulas, synthetic division, rational roots, multiplicity and bounds.
- Prove, once the fundamental theorem of algebra is admitted, the factorization of every real polynomial into linear factors and irreducible quadratic factors.
1. Division algorithm. Given two polynomials $P$ and $Q$ , with $Q$ nonzero, there exist a unique quotient $C$ and a unique remainder $R$ such that $P = Q \cdot C + R$ , with $R$ zero or of degree less than that of $Q$ .
  Rests on: A (Induction). Proved in: The algebra of real polynomials.
2. Remainder theorem and factor theorem. The remainder of dividing $P$ by $x - r$ is $P(r)$ ; therefore, $x - r$ divides $P$ if and only if $P(r) = 0$ .
  Uses: 10. Division algorithm. Proved in: The algebra of real polynomials.
3. Lemma on distinct roots. If $r_1, \ldots, r_k$ are distinct roots of $P$ , then $P = (x - r_1) \cdots (x - r_k) \, C$ ; consequently, a nonzero polynomial of degree $n$ has at most $n$ real roots.
  Uses: 2. Sign of a product, 11. Remainder theorem and factor theorem. Rests on: A (Induction). Proved in: The algebra of real polynomials.
4. Identity principle. Two polynomials take the same value at every real number if and only if they have the same coefficients, power by power; hence each polynomial is identified with its function, and the square root is not a rational function.
  Uses: 12. Lemma on distinct roots. Proved in: The algebra of real polynomials.
5. Quadratic formula and discriminant. By completing the square, the roots of $ax^2 + bx + c$ are $\frac{-b \pm \sqrt{\Delta}}{2a}$ , with $\Delta = b^2 - 4ac$ : there are two if $\Delta > 0$ , one if $\Delta = 0$ , and none if $\Delta < 0$ , in which case the polynomial is irreducible over the reals.
  Uses: 2. Sign of a product, 11. Remainder theorem and factor theorem. Proved in: Factoring quadratic polynomials and trinomials of degree 2n.
6. Factorization and Vieta’s formulas. If $\Delta \geq 0$ , then $ax^2 + bx + c = a(x - x_1)(x - x_2)$ , with $x_1 + x_2 = -\frac{b}{a}$ and $x_1 x_2 = \frac{c}{a}$ .
  Uses: 14. Quadratic formula and discriminant, 13. Identity principle. Proved in: Factoring quadratic polynomials and trinomials of degree 2n.
7. Biquadratic with negative discriminant. If $b^2 - 4ac < 0$ , the trinomial $ax^4 + bx^2 + c$ is $a$ times the product of two irreducible real quadratic polynomials, $x^2 - kx + m$ and $x^2 + kx + m$ .
  Uses: 14. Quadratic formula and discriminant. Proved in: Factoring quadratic polynomials and trinomials of degree 2n.
8. Synthetic division. The coefficients of the quotient of $P$ by $x - r$ are obtained by bringing down the leading coefficient and adding to each coefficient $r$ times the previous number; the last number is the remainder, $P(r)$ .
  Uses: 10. Division algorithm, 13. Identity principle. Proved in: Roots of higher-degree polynomials and synthetic division.
9. Rational roots. If the fraction in lowest terms $\frac{p}{q}$ is a root of a polynomial with integer coefficients with $a_0 \neq 0$ , then $p$ divides $a_0$ and $q$ divides $a_n$ ; the proof uses Euclid’s lemma from the first unit.
  Rests on: B (Euclid’s lemma). Proved in: Roots of higher-degree polynomials and synthetic division.
10. Repeated synthetic division. The root $r$ has multiplicity $k$ , that is, $P = (x - r)^k C$ with $C(r) \neq 0$ , if and only if, on dividing successively by $x - r$ , the first $k$ remainders are zero and the next one is not.
  Uses: 17. Synthetic division, 10. Division algorithm, 11. Remainder theorem and factor theorem. Proved in: Roots of higher-degree polynomials and synthetic division.
11. Bounds on the roots. Every real root satisfies $\abs{r} < 1 + \frac{M}{\abs{a_n}}$ (Cauchy bound), where $M$ is the largest of the absolute values of the other coefficients, and if synthetic division by $x - c$ , with $c > 0$ and $a_n > 0$ , gives a row without negative numbers, no real root exceeds $c$ .
  Uses: 17. Synthetic division. Proved in: Roots of higher-degree polynomials and synthetic division.
12. Factorization over the reals. Every real polynomial of degree $n \geq 1$ is its leading coefficient times factors $x - r$ , with $r$ real, and times monic quadratic factors irreducible over the reals.
  Uses: 14. Quadratic formula and discriminant, 10. Division algorithm, 17. Synthetic division. Rests on: A (Induction), E (Fundamental theorem of algebra), D (Conjugate roots). Proved in: Roots of higher-degree polynomials and synthetic division.
Segment 3 From the quotient to the graph
- Determine when a simplification is an equality and when two algebraic fractions are equal.
- Decompose a proper fraction into partial fractions.
- Read off from the factorization the sign of a polynomial or of a quotient and use it to solve inequalities and find domains.
- Analyze rational functions and functions with radicals: vertical, horizontal, oblique and curvilinear asymptotes, endpoints of the domain and range.
1. Simplification lemma. At every real number that does not make $QC$ vanish, the fraction $\frac{PC}{QC}$ takes the same value as $\frac{P}{Q}$ ; both are the same function only if the second is restricted to the domain of the first.
  Uses: 2. Sign of a product. Proved in: Algebraic fractions and partial fractions.
2. Equality of fractions. Two polynomials that coincide at all real numbers except finitely many are equal; it follows that $\frac{P}{Q}$ and $\frac{R}{S}$ coincide wherever $QS$ does not vanish if and only if $PS = QR$ .
  Uses: 12. Lemma on distinct roots, 13. Identity principle. Proved in: Algebraic fractions and partial fractions.
3. Partial fractions. Every proper fraction is, in a unique way, a sum of terms $\frac{A}{(x - r)^j}$ and $\frac{Bx + C}{(x^2 + px + q)^j}$ , one for each power of each factor of its denominator; with linear factors, existence is obtained by separating one term at a time and uniqueness by isolating at each root the highest power; with irreducible quadratic factors, both are admitted.
  Uses: 2. Sign of a product, 11. Remainder theorem and factor theorem, 10. Division algorithm, 23. Equality of fractions, 21. Factorization over the reals. Rests on: F (Bézout and unique factorization). Proved in: Algebraic fractions and partial fractions.
4. Sign of a polynomial. A product of $n$ nonzero factors is negative if it has an odd number of negative factors; therefore, the sign of a factored polynomial changes when crossing a root if and only if its multiplicity is odd, and the irreducible factors keep the sign of their leading coefficient.
  Uses: 2. Sign of a product, 14. Quadratic formula and discriminant, 19. Repeated synthetic division. Rests on: A (Induction). Proved in: Sign charts and inequalities.
5. Vertical asymptotes and removable points. If $r$ makes the denominator of $\frac{A}{B}$ vanish and $A(r) \neq 0$ , the line $x = r$ is a vertical asymptote; if $A(r) = 0$ , the common factor cancels and, if the simplified fraction is defined at $r$ , this is a removable point.
  Uses: 11. Remainder theorem and factor theorem, 22. Simplification lemma, 25. Sign of a polynomial. Rests on: G (Limits and continuity). Proved in: Sign charts and inequalities.
6. Asymptotes of a rational function. If $P = Q \cdot C + R$ , the difference between $\frac{P}{Q}$ and $C$ is $\frac{R}{Q}$ , which tends to $0$ at both ends: the quotient $C$ is the horizontal, oblique or curvilinear asymptote, and the sign chart decides in which direction the branches escape next to the vertical ones.
  Uses: 10. Division algorithm, 26. Vertical asymptotes and removable points. Rests on: G (Limits and continuity). Proved in: Rational functions: asymptotes and graphs.
7. Zeros and range of a rational function. By the intermediate value theorem, a rational function that changes sign on an interval of its domain vanishes on it, and one that on an interval of its domain takes values greater and smaller than any number takes there all real values; if the equation $y \cdot Q(x) = P(x)$ is of degree $2$ , its discriminant gives the range.
  Uses: 14. Quadratic formula and discriminant, 27. Asymptotes of a rational function. Rests on: G (Limits and continuity), H (Intermediate value). Proved in: Rational functions: asymptotes and graphs.
8. Domain and endpoints with radicals. The domain of a function with radicals is the intersection of the sets given by the sign charts of its conditions; if a root of odd multiplicity makes a radicand of the numerator vanish, the graph ends there on the $x$ -axis; if a denominator vanishes with a nonzero numerator, there is a vertical asymptote; and, by the intermediate value theorem, each branch takes all the values between those it reaches or approaches at its endpoints.
  Uses: 25. Sign of a polynomial, 26. Vertical asymptotes and removable points. Rests on: G (Limits and continuity), H (Intermediate value). Proved in: Functions with radicals: domain, signs and graph.
9. Asymptotes with radicals. Since $\sqrt{x^2} = \abs{x}$ , the asymptote of a function with radicals is found separately when $x \to \infty$ and when $x \to -\infty$ , and it may be different at each end; if $\frac{f(x)}{x}$ does not tend to a number, there is no linear asymptote.
  Uses: 27. Asymptotes of a rational function. Rests on: G (Limits and continuity). Proved in: Functions with radicals: domain, signs and graph.

Towards analytic geometry. The next unit studies curves of the plane through their equations: lines, the parabola, which is characterized by completing the square, the circle as the union of the graphs of $\pm \sqrt{r^2 - x^2}$ and the hyperbola $y = \frac{1}{x}$ with its two asymptotes.