Polynomials, Factoring, and Rational Expressions

This is Unit Two of College Algebra: polynomials, factoring, and rational expressions. The unit has one idea running through it, and the capstone states it outright at the end. Polynomials add, multiply, factor, and form fractions in the same way integers do — so this unit is the previous unit's arithmetic, replayed on objects with letters in them. That is worth holding onto, because the material is easy to experience as a long list of unrelated procedures. It is not. Nearly every rule here is a rule you already use on numbers, applied where the numbers have variables attached. Four entries are named errors, and all four come from the same instinct — that an operation reaching one part of an expression must reach the rest. It reaches factors and not terms, numerators and not denominators. Getting that boundary right is most of the unit. By the end you will be able to do five things. One: multiply two binomials, and recognise the two products worth knowing on sight. Two: say why squaring a sum produces a middle term. Three: run the factoring strategy in order, and check by multiplying back. Four: say why only factors cancel. Five: say why a fraction splits over its top and never over its bottom.

Three entries of definition. What qualifies, the number used to describe one, and the vocabulary for the small ones — and the condition on qualifying is stricter than it looks, ruling out things that otherwise resemble polynomials closely.

What qualifies.

Alg 061 Polynomials

A sum of terms, each a number multiplied by the variable raised to a whole-number power that is not negative. A negative power or a fractional one disqualifies the term outright.

The describing number.

Alg 062 Degree and Leading Coefficient

The largest power appearing anywhere in it. The term carrying that power is the leading one, and the number in front of it is the leading coefficient.

The names for small ones.

Alg 063 Monomials, Binomials, Trinomials

One term, two terms, three terms, each with its own name. The prefix is counting terms — it says nothing about how many different variables appear.

Two operations, and then a memory device. The device is worth putting in its place immediately: it is not a new rule, it is the distributive property in the one case where a name for the order helps.

Adding and subtracting.

Alg 065 Adding and Subtracting Polynomials

Terms with the same variable at the same power combine; the rest stay put. And subtracting hands its minus sign to every term behind it, not only the first one.

Multiplying.

Alg 067 Multiplying Polynomials

Every term of one factor multiplies every term of the other, and the results are added. Inside each product the numbers multiply and the exponents add — which is Unit One's product rule doing the work.

The memory device.

Alg 068 FOIL

An order for the four products when two two-term expressions multiply. It is the previous entry with a name attached, and it covers two-term expressions only — three terms and it stops applying.

Two products that appear constantly and are worth recognising without working them out. One of them produces a middle term and the other produces none — and an entry sits between them for the mistake of confusing which is which.

The first pattern.

Alg 070 Squaring a Binomial

Squaring a two-term sum gives the first squared, the last squared, and twice their product in between. That middle piece is what the pattern exists to remind you about.

And the error.

Alg 071 The Missing-Middle Error

Squaring each piece separately and stopping there, which loses the middle entirely. The multiplication is four products, and this error performs two of them.

The second pattern.

Alg 073 The Difference of Squares

Two expressions identical except for the sign between them, multiplied: the middle products cancel each other and only the two squares survive. The signs have to be opposite and the terms have to match.

Now the operation running backwards, and it comes with something unusual for a mathematics topic — an answer you can always check yourself, in a few seconds, without consulting anything.

The operation.

Alg 081 Factoring

Rewriting an expression as a product. And because multiplying the factors back has to return what you started with, every answer carries its own verification — one that fails the check is not an answer.

What to look for first.

Alg 082 The Greatest Common Factor

The biggest expression dividing evenly into every term. On any variable appearing throughout, take the smallest exponent it carries — that is the most every term can surrender.

Watch it come out.

Alg 083 Factoring Out the GCF

Identify the common factor, write it in front, and fill the brackets with what each term becomes after surrendering it. Every term must be able to give it up — one holdout means the candidate was too large.

Three-term expressions, with two methods depending on whether the leading coefficient is one. The second method also tells you something useful: when it fails, it has proved the thing cannot be factored with whole numbers.

The simple case.

Alg 085 Factoring x² + bx + c

Find two numbers whose product is the constant and whose sum is the middle coefficient, and they go straight into the brackets. That is the whole method when nothing sits in front of the squared term.

And the general case.

Alg 086 Factoring ax² + bx + c by Grouping

Find two numbers multiplying to the outer coefficients' product and adding to the middle, split the middle term into those two, and factor each half. If no whole-number pair exists, the expression does not factor over the integers — the search is also the test.

A pattern to recognise.

Alg 088 Factoring a Perfect Square Trinomial

Two ends that are perfect squares with twice their roots' product in between, folding back into a squared bracket. The ends alone prove nothing — the middle term is what confirms the pattern.

Three entries on two-term expressions, and the middle one is unusual: it exists to tell you that something does not work. Knowing which patterns have no factorisation saves as much time as knowing the ones that do.

The one that factors.

Alg 090 Factoring a Difference of Squares

Two perfect squares with a minus between them come apart into a matched pair of brackets with opposite signs. Both terms have to be perfect squares, and the sign has to be a minus.

The one that does not.

Alg 091 The Sum of Squares

Change that minus to a plus and nothing factors it over the real numbers — no pair of real-coefficient brackets multiplies back to it.

And the awkward pair.

Alg 093 The Sum and Difference of Cubes

Two cubes, added or subtracted, break into a two-term bracket and a three-term one. The sign to watch is in the middle of the longer bracket — it runs opposite to the sign in the short one.

One procedure that arranges everything so far into an order — and the reason an order is needed is that the wrong first move makes later moves invisible. The common factor always comes out first.

Watch the sequence run.

Alg 095 Choosing a Factoring Strategy

Pull out the common factor, count how many terms remain, match that count to the methods available for it, and multiply back to check. Finished means nothing left inside can still be factored — which is what completely is doing in the instruction.

A new object: one polynomial over another. It brings a genuinely new concern — values that break it — and then the rule governing cancellation, which is the single most misapplied rule in the entire course.

The object.

Alg 101 Rational Expressions

One polynomial divided by another. Wherever the bottom evaluates to zero the whole thing is undefined, and those values are excluded from consideration rather than quietly passed over.

The rule.

Alg 103 Only Factors Cancel

Cancelling removes something that multiplies the whole top and the whole bottom. The licence is that a thing divided by itself is one — and nothing else grants it.

The error.

Alg 104 The Canceled-Term Error

Crossing out pieces that appear on both sides of the bar without being factors. A three added on top and a three underneath do not cancel, and treating them as though they do changes the value.

The procedure.

Alg 106 Simplifying a Rational Expression

Factor the top completely, factor the bottom completely, and remove whatever whole factors they share. Factoring first is what turns terms into factors — cancel whole brackets, never pieces from inside one.

Three operations on these fractions, and they line up exactly with ordinary fractions. Two are straightforward. The third needs a common denominator, which is where the arithmetic actually lives.

Multiplying.

Alg 108 Multiplying Rational Expressions

Tops multiply and bottoms multiply. Factor and cancel before doing it, so the expressions stay small rather than expanding and then collapsing again.

Dividing.

Alg 109 Dividing Rational Expressions

Turn the second fraction upside down and multiply, which hands the problem to the operation you already have. Only the divisor flips.

What adding needs first.

Alg 111 The Least Common Denominator

The smallest expression both denominators divide into, built by taking every factor that appears in either and giving it the highest power it carries anywhere.

Watch adding run.

Alg 112 Adding and Subtracting Rational Expressions

Rewrite both over the common denominator, add the tops, and simplify what results. Each top gets multiplied by exactly what its own denominator was missing — no more than that.

One split that works and one that does not, and then the procedure where the invalid one gets attempted most often — because at that point in the working, the illegal move is exactly the shortcut you want.

The legal split.

Alg 114 Splitting a Fraction over Its Numerator

A sum on top of a fraction comes apart into separate fractions, with the same denominator serving each piece. The split runs across the top and only the top.

And the illegal one.

Alg 115 The Denominator-Split Error

Dividing across the terms underneath. A sum in the denominator is a single block and has to be treated whole — breaking it apart changes the value.

Where it gets attempted.

Alg 117 Simplifying a Complex Rational Expression

An expression with fractions inside its top or its bottom. Collapse each floor to a single fraction first, then divide. Flipping a floor that is still a sum is the previous error, arriving at the one moment it looks like progress.

The unit closes on the claim the opening made, now with everything behind it — and it adds a second observation about the error entries, which turn out to be arranged rather than scattered.

The capstone. The unit, as one claim.

Alg 120 The Arithmetic of Polynomials

Polynomials add, multiply, factor and form fractions the way integers do, so this unit was the previous one's arithmetic on new objects. And each named error stands guard at the point where the resemblance to ordinary numbers breaks down.

Five things you can now do. You can multiply two-term expressions and recognise the two products on sight — a square that carries a middle term, and a matched pair with opposite signs where the middle disappears. You can say where the middle term comes from: the multiplication is four products, and squaring term by term performs two of them. You can run the factoring strategy in order — common factor first, then count the terms — and check by multiplying back, which is a verification you always have. You can say why only factors cancel: the licence is a thing divided by itself being one, and a term that adds was never divided by anything. And you can say why a fraction splits over its top and never over its bottom — a summed denominator is one block, and breaking it changes the value.