Precalculus 2: Polynomials and rational functions

Explore polynomials as expressions of variables and coefficients using addition, multiplication, and subtraction. Define degree as the highest exponent, distinguish one- and two-variable polynomials, and preview basic operations.

Explore polynomials as functions of one or more variables, with coefficients and powers forming expressions denoted by p or q; note degree and domain (all real numbers).

Learn how zeros of polynomials are found through functional, algebraic, and graphical approaches, and how factoring reveals x-intercepts using the zero product property.

Factoring polynomials helps solve polynomial equations and inequalities and determines where polynomials are positive or negative. It aids graphing polynomials and rational functions and introduces partial fraction decomposition for integration.

Explore graphs of polynomials by analyzing domain, zeros (x-intercepts), and y-intercepts, then apply factoring, Cauchy’s bound, stacking factors method, and derivatives to study monotonicity.

Polynomials are continuous and differentiable, giving smooth graphs with no holes. Their derivatives, polynomials of lower degree, relate slope to monotonicity and mark maxima or minima where they vanish.

Explore why polynomials underpin many areas of mathematics, from Taylor polynomials and rational functions to differential equations and eigenvalues, highlighting simple computations and practical applications.

Explore rational functions as quotients of polynomials and determine their domain by factoring the denominator and excluding its zeros, illustrated by the example y equals 1/x.

Examine the similarities between integers and rational numbers and between polynomials and rational functions, including domains, arithmetic operations, division with remainder, and irreducible polynomials.

Contrast polynomials with integers by unveiling the fundamental theorem of algebra as a black box and introducing partial fraction decomposition as a gray box for rational functions.

Build the building blocks of polynomials by introducing powers with positive natural exponents, defining the base and exponent, and exploring sign rules and multiple ways to express numbers as powers.

Explore two product rules for powers: (ab)^n = a^n b^n and a^n b^m = a^{n+m}, with motivation and polynomial multiplication applications; includes practice exercises.

Explain why we need more rules for powers beyond positive natural exponents for polynomial multiplication. Introduce power and exponential functions with positive bases.

Learn two quotient rules for powers: (a/b)^n = a^n/b^n and (a^n)/(a^m) = a^{n-m}, with conditions n>m and nonzero bases, illustrated.

Apply the power rule to raise a power to another power by multiplying exponents, a key tool for polynomials and their compositions.

Define a^0 as 1 for all nonzero bases a, and justify it with the quotient rule for exponents, using n = m to show a^1/a^1 = a^0.

Define powers with negative integer exponents and show that a^(-m) equals 1 over a^m for nonzero a, using the quotient rule and the zero power rule.

Define powers with rational exponents and establish that a^(m/n) equals the nth root of a raised to m, with nonnegativity restrictions for even n.

Explore rules for computing powers with integer and rational exponents, distinguish undefined arithmetic results from indeterminate forms, and solve examples like 1/32, cube roots, and 8^(2/3) to reinforce power rules.

Review precedence rules for powers and exponents as this last video before polynomials. Point to precalculus prerequisites section 0.2 for extra exercises and the polynomial-relevant example linked to video 26.

Define univariate polynomials as expressions formed by addition and multiplication of real numbers and a variable. Identify coefficients, degree, leading term, and forms: monomial, binomial, trinomial, and the zero polynomial.

Identify the degree and coefficients of polynomials, determine monic status, and express non-monic polynomials as a product of a monic factor and a real number, using concrete examples.

Explore the domain of polynomial functions, show polynomials in one variable have domain real numbers, and contrast cases with square root terms, division, or sine terms that break polynomial form.

Evaluate polynomials by substituting values; p(0) equals constant term and p(1) equals the sum of coefficients, with p(-1) depending on degree parity; compare graphs with the dominating x^6 leading term.

Add and subtract polynomials by combining like terms, recognizing the same variables raised to the same powers, and applying the square of a difference formula for binomial expressions.

Multiply polynomials using the distributive law and scaling, reinforce polynomial multiplication, combine like terms, and apply power rules to expand products of polynomials.

Learn to prove and apply the difference of squares and the square of sum and difference formulas to factor fourth powers, including a^4−b^4 and a^4+b^4.

Master the composition of polynomials by applying inner and outer functions, and use Pascal's triangle and the binomial theorem to compute powers of sums.

Explore the composition of polynomials through concrete exercises, computing p circle q and q circle p at x=1. Learn to use coefficient sums and formulas to verify results.

Explain monic monomials x^n and their importance: graphs pass through the origin and (1,1), exhibit even/odd symmetry, and show leading-term domination for large x.

Explore how polynomial values behave at infinity, using monic monomials and leading terms to determine range for odd and even degrees, without calculus.

Analyze how scaling leaves degree unchanged, how product degrees add, how sums can cancel to lower degrees, and why the zero polynomial has degree minus infinity.

Explore linear equations and systems of linear equations, and learn how to identify linearity, rewrite into equivalent forms, and distinguish linear from non-linear equations.

Explore linear polynomials and straight lines: a linear polynomial p(x)=a1 x+a0 yields a straight line y=a1 x+a0, with zero x=-a0/a1, and two linked equations describing graphs and zeros.

Master practical methods for solving systems of linear equations used in calculus, including graphical and algebraic elimination techniques. Gaussian elimination is optional and may be explored for deeper understanding.

Solve for a, b, and c by translating the condition that three points lie on p(x)=ax^2+bx+c into a linear system, yielding the parabola p(x)=x^2-2x+3.

Translate the problem to a four-equation system to determine the cubic p(x) through four points; find a=1, b=3, c=0, d=-4, so p(x)=x^3+3x^2-4, verified at -1, 0, 1, 2.

Learn undetermined coefficients and ansatz to factor polynomials by guessing a second-degree q(x) with coefficients a, b, c, then equate coefficients to solve, yielding p(x) = (x-1)(x+2)^2.

Learn about overdetermined systems, where more equations than unknowns can be inconsistent or have a unique or infinite set of solutions. Verify a proposed solution satisfies all equations.

Show that the polynomial p is not divisible by d by assuming p equals d q and solving for a, b, c; the system is inconsistent.

Apply undetermined coefficients to partial fraction decomposition, determine constants a and b, and verify equality of polynomials via a common denominator.

Solve for a, b, and c in problem 6 using partial fraction decomposition, yielding a=1, b=-1, c=2, and revealing how equating numerators guides integration.

Shift from linear to quadratic by analyzing second-degree polynomials, their zeros and x-intercepts, and the standard form p(x)=ax^2+bx+c with factoring and parabola graphs.

This lecture presents the parabola y = x^2 as the most important, proving its increasing nature for x > 0, its even symmetry, and its nonnegative range.

Transform the standard parabola y = x^2 into other second-degree polynomials through translations, scalings, and reflections, noting vertex shifts and symmetry.

Master the method of completing the square to transform quadratics into a perfect square, locate a parabola’s vertex, and find x-intercepts by solving related equations.

Complete the square to find the vertex of a parabola and plot its graph, deriving vertex coordinates such as (-3, -49) and understanding the y-intercept.

Define the square root as the non-negative number whose square is the given value, so x^2 = 4 yields two roots, 2 and -2, via the difference of squares.

Learn to solve quadratic equations by completing the square to find zeros and vertices, using both the square root method and the difference of squares approach, and verify solutions by substitution.

Explore completing the square for factoring polynomials, linking zeros to factors, and applying to plotting parabolas, solving quadratics, and factoring second-degree polynomials.

Derive the quadratic formula from ax^2+bx+c via completing the square, compute the discriminant delta = b^2−4ac to find x1 and x2, and locate the vertex at x = −b/(2a).

Explore completing the square for quadratics with leading coefficients not equal to one and compare with the quadratic formula. Find roots, vertex, and the corresponding monic polynomial and parabola.

Explore that not all second-degree polynomials factor into two linear factors, examine parabola symmetry and zeros, and verify factorization by multiplication.

Discover Vieta's formulas, linking the sum and product of a quadratic's real zeros to its coefficients. See how monic and non-monic cases yield x1+x2 = -b/a and x1x2 = c/a.

Apply Vieta's formulas to solve two problems: compute x1^3+x2^3 from x1+x2 and x1x2 for a quadratic, and determine a from the ratio of its roots.

We confirm Vieta's formulas by deriving their sum and product from the quadratic formula for quadratics with nonzero a, showing the roots give −b/a and c/a.

Identify rational zeros of second-degree polynomials with integer coefficients using divisor rules: m divides the constant term and n divides the leading coefficient, with monic polynomials yielding only integers.

Extend the real numbers to complex numbers by introducing i with i^2 = -1, enabling addition, multiplication, and factoring polynomials via the fundamental theorem of algebra.

Explore complex numbers and their arithmetic, including conjugates, sums, products, and powers, with practice using binomial expansion and Pascal's triangle.

Explore solving quadratic equations with negative discriminants using complex numbers, by the quadratic formula and by completing the square, and confirm that roots are conjugate pairs.

Sign of a product equals sign of its quotient; division shares the same sign as multiplication by an inverse. Use factoring to solve polynomial and rational inequalities, noting domain restrictions.

Explore quadratic polynomials with zero coefficients, analyzing how coefficient a dictates parabola shape and zeros across four cases, including when b or c vanish.

Explore quadratic polynomials with all nonzero coefficients, using the vertex, the discriminant, and Vieta's formulas to classify parabolas by signs of a, b, and c, including eight cases.

Explore parabolas and second-degree polynomials through practice tests, comparing x^2 + bx + c = 0 and a x^2 + b x + c = 0 to determine real roots.

Use Vieta's formulas to analyze two real-root quadratics, x^2+px+7=0 and x^2+px-7=0, showing same-sign roots in the first and different-sign roots in the second; test one is indeterminate.

Apply vieta's formulas to the quadratic x^2 + b x + c = 0 with c nonzero, comparing same-sign and opposite-sign roots. The first test makes b positive; the second does not determine.

This lecture analyzes two test questions on a quadratic ax^2+bx+c that is negative for all real x, showing a and c must be negative and the discriminant determines the options.

Solve a quadratic equation in disguise by transforming x^4+4x^2+16=0 into quadratic factors, completing the square, and distinguishing real from complex solutions with conjugate roots.

Solve a quadratic equation in disguise by factoring and substituting y = x squared, then use Vieta's formulas to obtain five real roots: 0, ±√2, and ±√5.

Transform the quartic into a quadratic by grouping factors, substitute y = x^2+5x+4, solve y^2+2y-50=0, and obtain two real x and two complex roots via discriminants.

Factor the disguised quadratic by extracting x^2 to form two quadratics, x^2 - p x - q = 0 and x^2 - p x + q = 0. Use discriminants to find two real roots and two complex roots under q > p^2/4.

Explore factoring tricks for polynomials, including the square of a difference, the difference of squares, completing the square, PK formula, and the quadratic formula, with Pascal's triangle and rational roots.

Factor polynomials using sums or differences of powers and the difference of squares; apply the zero product property to find real roots, using discriminants and completing the square as needed.

We factor polynomials with Pascal's triangle binomial formulas to solve zero-product equations, identifying roots and multiplicities, including triple roots at -1 and 2.

Discover factoring by smart grouping to lower polynomial degree, solve real equations, and apply techniques like difference of squares, sum or difference of cubes, and completing the square.

Apply factoring by substitution to convert high-degree polynomials into quadratics using y = x^2 or y = x^3, solve with the quadratic formula, then back-substitute.

Learn to factor polynomials by grouping and substitution in a combined method, solving a quartic via y = x^2, a quadratic, and the discriminant, then back-substituting to x.

Explore the polynomial division theorem, compute quotient and remainder for polynomials with real coefficients, and apply degree rules to factor polynomials and solve equations.

Examine four methods of polynomial division, including long division, a multiplication-based approach using systems of equations, and Horner-Ruffini schemes for monic binomials, to find quotient and remainder.

Explore method one for polynomial division through example 1.1, linking the procedure to the theorem's proof. Apply induction to obtain quotient and remainder, with d(x)=x^2+2 and x^4 divided by d(x).

Perform long division on polynomials, obtaining the quotient and remainder. See how this method connects to the previous approach and reinforces the link between leading-term division and polynomial long division.

Explore the undetermined coefficients method for polynomial division through example 3.1, forming quotient and remainder by an ansatz, multiplying polynomials, and solving a linear system.

Verify polynomial division by multiplying the quotient with the divisor and adding the remainder to obtain the original polynomial, confirming the unique quotient and remainder for p and d.

Use the undetermined coefficients method to perform polynomial division, predict the quotient and remainder, and solve a coefficient system to satisfy p(x)=q(x)d(x)+r(x).

Apply method two of polynomial long division to example 2.2, deriving the quotient 2x^2 + 3x + 11 and the remainder 25x − 5.

Apply method one for polynomial division to compute p by d, dividing leading terms, compensating, and collecting like terms to yield quotient 2x^2+3x+11 and remainder 25x-5.

Practice and master polynomial division by solving four problems using undetermined coefficients and long division. Verify quotients and remainders, apply the remainder theorem, and study divisibility by x-1.

Explore why division by monic binomials is essential for factoring polynomials, using Horner/Ruffini's scheme and rational zeros to guide division and obtain a lower-degree quotient.

Explore the remainder theorem for polynomial division by x minus x0, showing the remainder equals p(x0) and introducing the factor theorem with quotient and remainder concepts.

Explore the factor theorem: identify zeros of polynomials, factor p(x) as (x - x0) q(x), and apply the remainder theorem to verify factors, enabling efficient factoring and solving.

Apply the remainder theorem to find remainders when dividing p by x-1 and x+2, by evaluating p at 1 and at -2. The results are -14 and 100, respectively.

Use the remainder theorem and the division theorem to determine the remainder of p divided by (x-1)(x-2), with p(1)=1 and p(2)=2, giving the remainder x.

Apply the remainder theorem to a polynomial with p(1)=1, p(2)=2, p(3)=3, dividing by (x-1)(x-2)(x-3). Solve for the quadratic remainder and find it equals x.

Use the remainder theorem to determine a, b, and c so the polynomial is divisible by x-1, x+2, and x-3, yielding minus seven, one, and six.

Explore polynomial division in action as you determine coefficients a and b so that p(x) divided by x^2+2 leaves a remainder x+1; obtain a=1 and b=3, yielding p(x)=x^5+3x^3+x^2+3x+3.

Learn polynomial division of p(x) by x^2 − 1 using smart grouping and undetermined coefficients, then compute p(1) and p(−1) to obtain r(x) = 1.

Explore the Ruffini-Horner scheme for dividing polynomials by monic binomials of first degree, derive the quotient and remainder, and connect to the synthetic division tableau.

Apply the Ruffini-Horner scheme to divide a degree four polynomial by x+1, fill a coefficient table, compute q(x)=3x^3-11x^2+18x-18, and obtain remainder 20.

Explore Ruffini–Horner scheme with problem 15 by dividing x^5 by x-1, building a six-column synthetic division table, obtaining quotient x^4+x^3+x^2+x+1 and remainder 1, noting zero coefficients.

Explore rational zeros of polynomials with integer coefficients using the polynomial division theorem and the factor theorem, and learn how divisors shape possible roots and factoring strategies.

Master factoring polynomials with integer coefficients by testing integer zeros, using division to reduce degree, and obtaining p(x) = (x+2)(x-1)(x-3) with zeros -2, 1, and 3.

Factor polynomials to solve problem 18 by testing rational roots, performing long division, and obtaining the factorization (x-1)(x-2)^2.

Test the divisors of 18 as potential rational zeros to factor the polynomial. Factor the polynomial into (x+3)(x+2)(x-1)(x-3) and note zeros at -3, -2, 1, and 3.

Factor the degree five polynomial using a known multiple zero at x=2, apply repeated division by x-2, reveal the factorization p(x)=(x-2)^3(x^2+x+1), and note the irreducible quadratic.

Explore Vieta's formulas for cubic polynomials, linking coefficients to roots, and see how the binomial theorem for cubes emerges as a special case.

Apply Vieta's formulas to construct a cubic with roots x1^2, x2^2, and x3^2; the coefficient-one form from problem 21 is x^3 - 6x^2 + 9x - 1 = 0.

Discover Cauchy’s bound for zeros of a polynomial: all zeros lie in [-m-1, m+1], where m is the largest of the quotients |a_i|/|a_n| of the coefficients.

Explore the fundamental theorem of algebra and its consequences, showing that every real-coefficient polynomial factors over the complex numbers into linear factors, with complex zeros and multiplicities.

See how polynomials with real coefficients factor into real linear factors for real zeros and irreducible quadratics for complex conjugate pairs, guided by complex conjugates and discriminants.

Explore how polynomials with real coefficients have a parity-based number of zeros, determined by degree and multiplicities, via factorization into real linear factors or irreducible quadratic factors.

Investigate factorization of a tough polynomial by positing a quadratic and a cubic, applying the rational zeros test and undetermined coefficients to obtain a quadratic irreducible factor and a cubic.

Factor polynomials over R and C using the rational root test and division. Obtain zeros at 2 and -3, yielding q(x)=x^2-6x+10, irreducible over R but factors over C as (x-3-i)(x-3+i).

Learn to factor polynomials with real coefficients over real and complex numbers, using complex conjugate zeros and quadratic factors to obtain linear factors.

Explore factorization of polynomials over real and complex numbers. Learn how a known complex zero with real coefficients implies a conjugate zero and enables factorization into linear factors.

Explore how real polynomials can defy school expectations, moving beyond integer or rational coefficients, rational zeros, and division-based factoring, as you learn to construct polynomials with specified properties.

Construct a degree-five polynomial with a single real root using a linear factor x minus sqrt(17) and irreducible quadratics, exploring deliberate complexity and discriminant ideas.

Construct a degree seven polynomial with exactly three real roots using three linear factors, then extend with higher-degree factors to reach seven while exploring irreducible forms and negative discriminants.

Learn to construct an eight-degree polynomial with no real roots by combining quadratic factors like (x^2+1)^4 and others with negative discriminants, illustrating infinite solutions.

Explore what constitutes a polynomial equation and a polynomial inequality by comparing two polynomials p1 and p2, and identify the x values that satisfy equality or a given inequality.

Explore inequalities you can solve without computations by analyzing polynomials and real solutions. See when even exponents with positive coefficients guarantee positivity, and that positive coefficients alone may yield zeros.

Explore functional equations and inequalities through graphical illustrations of two polynomials f and g, identifying where g lies above f and using h(x)=f(x)-g(x) zeros to locate intersections.

Graph f and line y = x + 3 to solve x^2 + x - 6 > x + 3, locate intersections at -3 and 3 to define solution set.

Visualize graphically the polynomial inequality by comparing f(x)=a-x^2-x+6 with g(x)=x+2, finding their intersection points and the open solution interval.

Explore how to solve a polynomial equation and its inequalities graphically and computationally by intersecting two parabolas, yielding x=4, and use h(x)=2x−8 to determine x>4 or x<4.

Explore graphical and computational methods for solving a polynomial equation between two polynomials, find intersections at x = -1 and x = 1, and determine inequality solutions via graphs.

Visualize two third-degree polynomials, examine their difference, and locate intersection points, excluding them; state the solution as the union of intervals where the blue graph lies above the rose graph.

Explore how to solve polynomial equations by subtracting polynomials, factoring, and using zeros. See how polynomial inequalities extend with sign analysis and monic binomial factors for roots.

Use the stacking factors method with a table to solve polynomial inequalities by analyzing signs across intervals divided by zeros -2, 1, and 3, determining where p(x)=0, p(x)>0, or p(x)<0.

Apply the stacking factor method in a coordinate system to analyze polynomial inequalities by partitioning domain at zeros, evaluating the signs of factors on each interval, and deducing graph.

Learn a graphical method for solving polynomial inequalities using monic monomials and irreducible factorization, and apply odd versus even multiplicities to identify sign changes.

Solve a fourth-degree polynomial inequality using both table and graph methods; factor into (x+2)(x+1)(x-1)(x-2), analyze signs across roots -2, -1, 1, 2, and obtain the solution set (-infinity, -2] ∪ [-1, 1] ∪ [2, infinity).

Solve the inequality using both table and graphical methods after factoring the polynomial (via quadratic formula and completing the square), identifying zeros at -3, 1/2, 1, and 2.

Apply the quick graph method to solve polynomial inequalities by factoring into irreducibles, discarding nonessential factors, and reading off solution intervals like x<1, 3<x<5, and -8 <= x <= 7.

Apply the quick graph method to solve four precalculus polynomials and rational functions inequalities by factoring and locating zeros, determining sign intervals and solution sets.

Solve a degree six polynomial inequality by factoring by hand using horner's method, identify zeros at x = -3, 1, 3, factor to (x+3)^2(x-1)^3(x-3), and apply the quick graph method.

Apply the quick graph method to solve an inequality from an unfactored polynomial by finding zeros, factoring, and testing intervals to yield -2 to 1 and 3 to infinity.

Critiquing sign diagrams, the lecture explains polynomial inequalities and contrasts methods revealing behavior across intervals, including test values, and urges you to choose an approach ensuring understanding and accuracy.

Construct a fifth-degree polynomial with zeros at -3, 3, -2, and 4, with a negative leading term, and show -2 is a double zero that yields a local maximum.

Explore how continuity ties limits to function values for polynomials, using the epsilon-delta idea. A function is continuous at a if the limit as x approaches a equals f(a).

The lecture motivates that all polynomials are continuous functions, building from the identity function and using epsilon-delta continuity, then showing that sums and scalings of continuous functions remain continuous.

Polynomials satisfy the naive idea that continuity means drawing without lifting a pen, while floor functions on all reals expose limits of that idea, requiring epsilon-delta criteria and domain awareness.

Learn the derivative as the tangent slope and the rate of change of polynomials; use secant lines and limits to locate increasing intervals and local/global maxima and minima.

Learn how derivatives of polynomials form a new polynomial by differentiating monomials and sums, with constants yielding zero, and use the derivative to analyze monotonicity and extrema.

Return to monic monomials to analyze their behavior around zero and for large x. Formalize the limit behavior as x tends to infinity for positive integers, and introduce multiplicative inverses.

Formalize infinite limits as x approaches infinity or minus infinity with an epsilon-delta framework using large E and delta, applied to polynomials and quadrants to show unbounded growth of f(x).

Demonstrate finite limits at infinity for rational functions using epsilon-delta definitions and visualize how f(x) approaches a horizontal asymptote.

Define infinite limits at a point for rational functions using an epsilon-delta criterion: as x approaches a, f(x) exceeds any bound, with possible right- or left-hand infinity.

Explore the types of limits and where to find them, and learn to define them with epsilon and delta. Compare point limits and infinity limits for polynomials and rational functions.

This lecture explains graphs of polynomials, covering domain, range for odd and even degrees, zeros, y-intercept, and how the derivative guides increasing and decreasing behavior.

Explore how the graph of polynomials relates to their derivatives, revealing monotonicity, local maxima and minima, and the roles of derivative zeros in shaping the curve.

Sketch the cubic polynomial p(x)=x(x-2)^2 by factoring to locate zeros at 0 and 2 (double), find the y-intercept, then use derivatives to identify critical points and compare with software.

Factor the cubic p(x)=x^3-3x-2 to find zeros at -1 (double) and 2, use p'(x)=3(x-1)(x+1) to locate critical points, and sketch its end behavior and y-intercept.

Explore graphs of fourth-degree polynomials, identify zeros, and use derivatives to refine sketches, including rational and irrational zeros, factoring, and y-intercept analysis.