Polynomials and Rational Expressions Study Guide: 78 Flashcards from FOIL to Abel-Ruffini
Study polynomials and rational expressions with 78 flashcards: degree, factoring, synthetic division, remainder theorem, multiplicity and partial fractions.
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Polynomials look like a vocabulary chapter until the first factoring quiz. Then you discover that you need the vocabulary and the moves. The Polynomials & Rational Expressions deck (code JE8840, 78 cards) gives you mostly the vocabulary, which tells you how to study it.
Study this deck: Polynomials & Rational Expressions
1 / 10Polynomial
Tap the card for the answer · swipe right if you knew it, left if not
What the 78 cards cover
Monomials, binomials, trinomials and multinomials; coefficients, indeterminates, constant terms and the leading coefficient; degree of a term vs degree of a polynomial (and why the zero polynomial's degree is undefined, −1 or −∞); linear, quadratic, cubic, quartic and quintic; univariate, bivariate and multivariate; like terms and standard form; FOIL; long division, synthetic division and the remainder theorem; GCF, difference of squares and irreducible polynomials; roots, multiplicity and the Fundamental Theorem of Algebra; Horner's method; rational expressions, proper vs improper, partial fractions; the binomial theorem; Cardano, Ferrari and the Abel–Ruffini theorem.
Sample cards
| Front | Back |
|---|---|
| Degree of a polynomial | The largest degree of any term with a nonzero coefficient |
| Degree of the zero polynomial | Undefined (or defined as negative: −1 or −∞) |
| Multiply (x + 2)(x + 3) | x² + 5x + 6 |
| Factor x² + 5x + 6 | (x + 2)(x + 3) |
| Factor 6x³ + 9x² | 3x²(2x + 3) |
| Polynomial remainder theorem | The remainder when dividing P(x) by (x − c) is P(c) |
| A root has multiplicity 2 | (x − r)² is a factor; the root r is repeated twice |
| Improper rational expression | Numerator degree ≥ denominator degree |
| Can quintic equations be solved by radicals? | No, in general (Abel–Ruffini theorem) |
How to study it
Count the cards that calculate. Only 9 of the 78 ask you to actually do something (cards 27, 30, 41, 43, 45, 54, 59, 74, 76). The rest are definitions. So review the deck as a glossary, and next to it write your own drill cards for the procedures it only names: one synthetic division by (x − 2), one remainder-theorem check (P(2) should equal that remainder), one long division leaving a remainder, one partial fraction split such as 1/(x² − 1), one (a + b)³ expansion. Five cards like this are worth more than twenty extra definitions.
Pair the mirror cards. Card 30 multiplies (x + 2)(x + 3), and card 43 factors the result back. Study them as a pair: factoring is FOIL run in reverse.
Repair two cards. Card 59 is missing its "Simplify" (it reads "the rational expression (x² − 4)/(x − 2)"). Its answer x + 2 also needs the condition x ≠ 2. Cancelling a factor never removes the hole in the graph, and teachers take points for this. Card 3 has lost "Etymology of" in the same way. Also note that the deck never states the quadratic formula itself. Card 69 only says what it is used for, so add x = (−b ± √(b² − 4ac)) / 2a.
Skip or keep the calculus. Cards 53–55 (derivatives and antiderivatives of xⁿ) belong to calculus. If you are in an algebra course, suspend them.
Keep going in Algebra
Quadratics come right after this deck. Master Quadratic Equations and Linear Systems covers the discriminant and the formula that card 69 leaves out.
Find deck JE8840 in the Flip It library and start with the 9 cards that calculate.
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