Polynomial and Rational Functions

This is Unit Seven of College Algebra: polynomial and rational functions. Unit Six read everything about a parabola from its formula. This unit does the same job on harder functions, and the capstone states the law that makes it possible: the factors carry the behaviour. That is why nearly every procedure here begins with factoring. Where a graph crosses, where it merely touches, where it runs off to infinity, and where it has a single missing point — all of it is read out of the factored form, and none of it is visible before you factor. One error entry sits at the centre of the unit and it is worth flagging now. Cancelling a factor from a fraction simplifies the expression and does not restore the input that factor excluded. The value stays out. Forgetting that is how a graph acquires a point it should not have. By the end you will be able to do five things. One: read a polynomial's arms from its leading term alone. Two: say what a repeated factor does to the graph at that point. Three: say why cancelling never restores an excluded value. Four: find a horizontal asymptote by comparing degrees. Five: write a variation equation from a description.

Three entries, and the third does real work. A polynomial can have any number of terms, and its behaviour far out at both ends is settled entirely by one of them — everything else stops mattering once the inputs get large enough.

The simplest family.

Alg 351 Power Functions

A single variable power carrying a single number in front, with nothing else added on. The building block for everything in the topic.

What the arms do.

Alg 352 End Behavior

Where the outputs head as inputs run far out in each direction. Two arms, described independently.

And where they come from.

Alg 354 End Behavior from the Leading Term

The arms match the power function in the highest-degree term. An even degree sends both arms the same way; an odd degree sends them opposite ways; and the leading coefficient's sign decides which way.

Two entries about the middle of the graph. The first explains something that puzzles people — why a graph sometimes touches the axis and turns back rather than passing through. The answer is an exponent.

Cross, or touch.

Alg 356 Zeros and Multiplicity

The power on a factor in the factored form decides the behaviour there. An odd power sends the graph through the axis; an even one has it touch and turn back without crossing.

How many turns.

Alg 358 Turning Points

The places where rising becomes falling or the reverse, and there can never be more than one fewer than the degree. A ceiling rather than a count.

One procedure assembling the last two topics, with a rule attached about what these graphs never do — and the entry calls it doctrine, which is the right word, because it is a fact about the whole family rather than a drawing tip.

Watch one get sketched.

Alg 360 Sketching a Polynomial

Set the arms from the leading term, place the zeros with their crossing or touching behaviour, mark where it meets the vertical axis, and join smoothly. Smoothly is required — these graphs have no corners, no breaks, and no gaps.

Two division procedures. One works on anything and one is much faster with a narrow licence, and both carry a bookkeeping warning — the fast one about which number to use, the general one about columns that are easy to lose.

The general algorithm.

Alg 364 Polynomial Long Division

The familiar arithmetic procedure applied to terms — divide the leading pieces, multiply back, subtract, repeat. Any missing degree has to be written in as a zero term, or the columns misalign and the subtraction goes wrong.

The shortcut.

Alg 366 Synthetic Division

Bring down, multiply, add across, and read off the quotient and remainder — using coefficients alone. It applies only to divisors of one particular shape, and the number used is the zero of that divisor rather than the number written in it.

Two results that together turn division into a zero-finding tool. The first says division is secretly evaluation; the second says a zero remainder means a factor. Combine them and one known zero unlocks the rest.

Division as evaluation.

Alg 368 The Remainder Theorem

Dividing by a linear factor leaves a remainder equal to the function's value at that number. The division computed something you could have got by substituting.

Zero and factor.

Alg 369 The Factor Theorem

A number is a zero of the polynomial exactly when the matching linear expression divides it evenly. The two statements are one fact in two vocabularies.

Using them together.

Alg 371 Finding Zeros with the Factor Theorem

Take one known zero, divide its factor out, and solve the smaller polynomial that remains. Confirm the starting zero before building on it — a single division with a zero remainder is that confirmation.

A new family, and the interesting thing is that a denominator zero produces two completely different results depending on whether its factor survives cancelling. Two behaviours, one condition telling them apart, and an error entry sitting between them.

The family.

Alg 376 Rational Functions

One polynomial over another. Every input is allowed except the ones making the bottom zero, and that exclusion is the family's defining feature.

The first behaviour.

Alg 377 Vertical Asymptotes

Where a denominator zero's factor does not cancel, the graph races off to infinity alongside a vertical line. The factor surviving is what produces it.

The second.

Alg 378 Removable Discontinuities

Where the factor does cancel, there is no asymptote — only one point missing from an otherwise ordinary graph. And the value stays excluded from the domain regardless.

The error.

Alg 379 The Erased-Hole Error

Treating the simplified expression as having the same domain as the original. Cancelling changes how the expression is written and cannot change what was never allowed in — the hole does not close.

One entry on behaviour far out, and it is decided by a comparison rather than a calculation — which of the two polynomials has the higher degree, with three possible outcomes.

Three cases.

Alg 381 Horizontal Asymptotes

A heavier bottom sends outputs toward zero. Equal degrees settle on the ratio of the leading coefficients. A heavier top means no horizontal asymptote at all — the outputs keep growing.

One procedure gathering every feature, and its instruction is the unit's law stated as a step: factor both floors before reading anything, because none of the features are visible until you do.

Every feature, in order.

Alg 383 Analyzing a Rational Function

Factor the top and the bottom, then read the domain, the holes, the vertical asymptotes, and the horizontal behaviour off the factors. Every feature lives in the factored form, so the order is not a preference.

Two entries. A third family with a domain restriction, and then a technique for giving an inverse to a function that failed Unit Five's reversibility test — by cutting the function down until it passes.

The third family.

Alg 386 Radical Functions and Their Domains

The variable sitting under a radical. With an even index the domain is wherever the inside is not negative, which is usually the whole question.

Earning an inverse.

Alg 388 Restricting a Domain for an Inverse

A function failing the one-to-one test can still be inverted on part of itself — choose a stretch where no output appears twice, and invert that stretch. The inverse belongs to the piece rather than the original.

Three relationships described in ordinary words, and a procedure for turning the words into equations. The middle one shares a word with something from Unit Five that it has nothing to do with, and the entry says so.

The first.

Alg 391 Direct Variation

One quantity a constant multiple of another, so they rise and fall together in step. The constant is what the problem usually asks you to find.

The second.

Alg 392 Inverse Variation

One quantity a constant divided by another, so increasing one decreases the other. Note the vocabulary collision — this is a relationship between quantities and has nothing to do with reversing a function.

The third.

Alg 393 Joint Variation

One quantity varying directly with two others multiplied together — the same idea as the first, with a product where a single variable stood.

Turning words into an equation.

Alg 395 Solving a Variation Problem

Read which relationship the description states, write that equation, use the given pair to find the constant, then answer. Match the form to the words first — the two main forms read similarly and produce opposite behaviour.

The unit closes on the law that has been running underneath every procedure — and stating it as a law rather than a habit explains why the factoring step was never optional.

The capstone. One law.

Alg 410 The Anatomy of Higher Functions

Polynomial and rational functions are read the way parabolas were — every behaviour taken from the formula — and the law making it work is that the factors carry it. Every claim here carries an earlier number.

Five things you can now do. You can read a polynomial's arms from its leading term alone — even degree sends them the same way, odd degree opposite ways, and the leading sign decides which. You can say what a repeated factor does at a zero: an odd power sends the graph through the axis and an even one has it touch and turn back. You can say why cancelling never restores an excluded value — the writing changed and what was never allowed in is still not allowed in. You can find a horizontal asymptote by comparing degrees: heavier bottom goes to zero, equal degrees give the leading ratio, heavier top gives none at all. And you can turn a description into a variation equation, matching the form to the words before touching any numbers.