Algebra Flashcards: Rewriting Rational Expressions

What this deck covers

This deck focuses on Rewriting Rational Expressions, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is $\frac{x^3+1}{x^2-x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+1$. Factor $x^3+1=(x+1)(x^2-x+1)$ then cancel denominator.

Flashcard 2: What is $\frac{x^2+1}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+\frac{1}{x}$. Split each term: $\frac{x^2}{x}+\frac{1}{x}=x+\frac{1}{x}$.

Flashcard 3: What is $\frac{x^2-1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x-1$. Factor as $(x+1)(x-1)$ then cancel $(x+1)$.

Flashcard 4: What is $\frac{x^3-1}{x^2+x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x-1$. Factor $x^3-1=(x-1)(x^2+x+1)$ then cancel denominator.

Flashcard 5: What is $\frac{x^3+x^2-x-1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2-1$. Factor and divide: $(x+1)(x^2-1)\div(x+1)=x^2-1$.

Flashcard 6: What is $\frac{x^3+2x+1}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+\frac{2x+1}{x^2}$. Split first term then combine remainder: $x+\frac{2x+1}{x^2}$.

Flashcard 7: What is $\frac{x^2+1}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+\frac{1}{x}$. Split each term: $\frac{x^2}{x}+\frac{1}{x}=x+\frac{1}{x}$.

Flashcard 8: What is $\frac{x^3-3x+2}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2-3+\frac{2}{x}$. Split each term: $\frac{x^3}{x}-\frac{3x}{x}+\frac{2}{x}$.

Flashcard 9: What is $\frac{x^3+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2-x+1$. Factor $x^3+1=(x+1)(x^2-x+1)$ then cancel $(x+1)$.

Flashcard 10: What is $\frac{x^3+2}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+\frac{2}{x}$. Split each term: $\frac{x^3}{x}+\frac{2}{x}=x^2+\frac{2}{x}$.

Flashcard 11: What is $\frac{x^3-x^2+x-1}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+1$. Factor and divide: $(x-1)(x^2+1)\div(x-1)=x^2+1$.

Flashcard 12: What is $\frac{x^3+2x^2-5x+6}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+2x-5+\frac{6}{x}$. Split each term: $x^2+2x-5+\frac{6}{x}$.

Flashcard 13: What is $\frac{x^2+4x+7}{x+2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+2+\frac{3}{x+2}$. Long division gives quotient $x+2$ with remainder $3$.

Flashcard 14: What is $\frac{x^3-2x^2+4}{x-2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+\frac{4}{x-2}$. Long division gives quotient $x^2$ with remainder $4$.

Flashcard 15: What is $\frac{x^3+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2-x+1$. Factor $x^3+1=(x+1)(x^2-x+1)$ then cancel $(x+1)$.

Flashcard 16: What is $\frac{2x^3+3x^2+x+5}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $2x+3+\frac{-x+2}{x^2+1}$. Long division gives quotient $2x+3$ with remainder $-x+2$.

Flashcard 17: What is $\frac{x^3+x^2+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+\frac{x^2-x+1}{x^2+1}$. Long division gives quotient $x$ with remainder $x^2-x+1$.

Flashcard 18: What is $\frac{x^2-4x+1}{x-2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x-2-\frac{3}{x-2}$. Long division gives quotient $x-2$ with remainder $-3$.

Flashcard 19: Identify the missing term form needed to divide $2x^4-x^2+5$ by $x^2+1$ using long division.

Answer: Rewrite as $2x^4+0x^3-x^2+0x+5$. Add zero coefficients for missing $x^3$ and $x$ terms.

Flashcard 20: What is $\frac{x^3+2x}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+2$. Split each term: $\frac{x^3}{x}+\frac{2x}{x}=x^2+2$.

Flashcard 21: What is $\frac{x^3+2x^2+3}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+2+\frac{3}{x^2}$. Split each term: $\frac{x^3}{x^2}+\frac{2x^2}{x^2}+\frac{3}{x^2}$.

Flashcard 22: What is $\frac{x^3+1}{x^2-x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+1$. Factor $x^3+1=(x+1)(x^2-x+1)$ then cancel denominator.

Flashcard 23: Identify the missing term form needed to divide $2x^4-x^2+5$ by $x^2+1$ using long division.

Answer: Rewrite as $2x^4+0x^3-x^2+0x+5$. Add zero coefficients for missing $x^3$ and $x$ terms.

Flashcard 24: What is $\frac{2x^3+3x^2+x+5}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $2x+3+\frac{-x+2}{x^2+1}$. Long division gives quotient $2x+3$ with remainder $-x+2$.

Flashcard 25: What is $\frac{3x^2-2x+4}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $3x+1+\frac{5}{x-1}$. Long division gives quotient $3x+1$ with remainder $5$.

Flashcard 26: What is the remainder degree condition when dividing by a linear $b(x)$ with $\deg(b(x))=1$?

Answer: $r(x)$ is a constant (degree $0$). Linear divisor means remainder has degree less than 1.

Flashcard 27: What is $\frac{4x^2+1}{2x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $2x+\frac{1}{2x}$. Split each term: $\frac{4x^2}{2x}+\frac{1}{2x}=2x+\frac{1}{2x}$.

Flashcard 28: What is $\frac{x^2+3x+2}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+3+\frac{2}{x}$. Split each term: $\frac{x^2}{x}+\frac{3x}{x}+\frac{2}{x}$.

Flashcard 29: What is $\frac{x^3+2x^2-5x+6}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+2x-5+\frac{6}{x}$. Split each term: $x^2+2x-5+\frac{6}{x}$.

Flashcard 30: What is $\frac{x^2+5x+6}{x+2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+3$. Factor as $(x+2)(x+3)$ then cancel $(x+2)$.

Flashcard 31: What is the remainder degree condition when dividing by a quadratic $b(x)$ with $\deg(b(x))=2$?

Answer: $r(x)$ has form $ax+b$. Quadratic divisor means remainder has degree less than 2.

Flashcard 32: What is $\frac{x^3+2x}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+2$. Split each term: $\frac{x^3}{x}+\frac{2x}{x}=x^2+2$.

Flashcard 33: What is the name of the process used to rewrite $\frac{a(x)}{b(x)}$ as $q(x)+\frac{r(x)}{b(x)}$?

Answer: Polynomial long division. The systematic method for dividing polynomials with remainder.

Flashcard 34: What is $\frac{x^3-2x^2+4}{x-2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+\frac{4}{x-2}$. Long division gives quotient $x^2$ with remainder $4$.

Flashcard 35: What is the rewritten form of $\frac{a(x)}{b(x)}$ after division with remainder?

Answer: $\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$. Standard quotient-plus-remainder form after polynomial division.

Flashcard 36: What is $\frac{x^3+3x^2+3x+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+2x+1$. Recognize as $(x+1)^3$ expanded then divide by $(x+1)$.

Flashcard 37: What is $\frac{x^2-1}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+1$. Factor as $(x-1)(x+1)$ then cancel $(x-1)$.

Flashcard 38: What is $\frac{x^2+5x+5}{x+2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+3-\frac{1}{x+2}$. Long division gives quotient $x+3$ with remainder $-1$.

Flashcard 39: What is $\frac{x^2-4}{x-2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+2$. Factor as $(x-2)(x+2)$ then cancel $(x-2)$.

Flashcard 40: What is $\frac{x^2+2x+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+1$. Factor as $(x+1)^2$ then cancel $(x+1)$.

Flashcard 41: What is $\frac{x^3+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+\frac{-x+1}{x^2+1}$. Long division gives quotient $x$ with remainder $-x+1$.

Flashcard 42: What is $\frac{x^4+1}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+\frac{1}{x^2}$. Split each term: $\frac{x^4}{x^2}+\frac{1}{x^2}=x^2+\frac{1}{x^2}$.

Flashcard 43: What is $\frac{3x^2+6}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $3x+\frac{6}{x}$. Split each term: $\frac{3x^2}{x}+\frac{6}{x}=3x+\frac{6}{x}$.

Flashcard 44: What is $\frac{x^4-x}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^3-1$. Factor out $x$: $\frac{x(x^3-1)}{x}=x^3-1$.

Flashcard 45: What is the name of the process used to rewrite $\frac{a(x)}{b(x)}$ as $q(x)+\frac{r(x)}{b(x)}$?

Answer: Polynomial long division. The systematic method for dividing polynomials with remainder.

Flashcard 46: What is $\frac{x^3+4x^2}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+4$. Split each term: $\frac{x^3}{x^2}+\frac{4x^2}{x^2}=x+4$.

Flashcard 47: What is $\frac{x^2+2x+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+1$. Factor as $(x+1)^2$ then cancel $(x+1)$.

Flashcard 48: What is $\frac{x^2+5x+6}{x+2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+3$. Factor as $(x+2)(x+3)$ then cancel $(x+2)$.

Flashcard 49: What is $\frac{4x^2+1}{2x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $2x+\frac{1}{2x}$. Split each term: $\frac{4x^2}{2x}+\frac{1}{2x}=2x+\frac{1}{2x}$.

Flashcard 50: Identify the missing term form needed to divide $x^3+2x+1$ by $x+1$ using long division.

Answer: Rewrite as $x^3+0x^2+2x+1$. Add zero coefficient for missing $x^2$ term.

Flashcard 51: What is $\frac{x^3+2x^2+x+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+2+\frac{x-1}{x^2+1}$. Long division gives quotient $x+2$ with remainder $x-1$.

Flashcard 52: What is $\frac{x^3+x^2+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+\frac{x^2-x+1}{x^2+1}$. Long division gives quotient $x$ with remainder $x^2-x+1$.

Flashcard 53: What is $\frac{x^3+3x^2+3x+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+2x+1$. Recognize as $(x+1)^3$ expanded then divide by $(x+1)$.

Flashcard 54: What is $\frac{x^3-3x+2}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2-3+\frac{2}{x}$. Split each term: $\frac{x^3}{x}-\frac{3x}{x}+\frac{2}{x}$.

Flashcard 55: What is $\frac{x^3-x}{x^2-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x$. Factor as $x(x^2-1)=x(x+1)(x-1)$ then simplify.

Flashcard 56: What is $\frac{2x^2+3x+4}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $2x+1+\frac{3}{x+1}$. Long division gives quotient $2x+1$ with remainder $3$.

Flashcard 57: What is $\frac{2x^2+3x+4}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $2x+1+\frac{3}{x+1}$. Long division gives quotient $2x+1$ with remainder $3$.

Flashcard 58: What is the required relationship between degrees in $\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$?

Answer: $\deg(r(x))<\deg(b(x))$. The remainder degree must be less than the divisor degree.

Flashcard 59: What is $q(x)$ called in $a(x)=b(x)q(x)+r(x)$?

Answer: The quotient polynomial $q(x)$. The result when $a(x)$ is divided by $b(x)$.

Flashcard 60: What is $\frac{x^3+2x^2+x+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+2+\frac{x-1}{x^2+1}$. Long division gives quotient $x+2$ with remainder $x-1$.

Flashcard 61: What is $r(x)$ called in $a(x)=b(x)q(x)+r(x)$?

Answer: The remainder polynomial $r(x)$. What's left after division when $\deg(r)<\deg(b)$.

Flashcard 62: What is the remainder degree condition when dividing by a linear $b(x)$ with $\deg(b(x))=1$?

Answer: $r(x)$ is a constant (degree $0$). Linear divisor means remainder has degree less than 1.

Flashcard 63: What is $\frac{x^3-1}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2+x+1$. Factor $x^3-1=(x-1)(x^2+x+1)$ then cancel $(x-1)$.

Flashcard 64: What is the key stopping condition in polynomial long division?

Answer: Stop when $\deg(\text{remainder})<\deg(b(x))$. Division ends when remainder degree drops below divisor degree.

Flashcard 65: What is $\frac{x^2+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x-1+\frac{2}{x+1}$. Long division gives quotient $x-1$ with remainder $2$.

Flashcard 66: What must you do first before long dividing by $x^2+3$ if terms are missing in $a(x)$?

Answer: Insert missing powers with coefficient $0$. Ensures all terms are present for proper column alignment.

Flashcard 67: What is $\frac{2x^2+3x+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $2x+1$. Factor as $(x+1)(2x+1)$ then cancel $(x+1)$.

Flashcard 68: What is $\frac{x^2-4}{x-2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+2$. Factor as $(x-2)(x+2)$ then cancel $(x-2)$.

Flashcard 69: What is $\frac{x^3-x}{x^2-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x$. Factor as $x(x^2-1)=x(x+1)(x-1)$ then simplify.

Flashcard 70: What is $\frac{3x^2-2x+4}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $3x+1+\frac{5}{x-1}$. Long division gives quotient $3x+1$ with remainder $5$.

Flashcard 71: What is $\frac{5x^3-10x}{5x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x^2-2$. Split each term: $\frac{5x^3}{5x}-\frac{10x}{5x}=x^2-2$.

Flashcard 72: What is the identity that connects division form and remainder form for polynomials?

Answer: $a(x)=b(x)q(x)+r(x)$. The fundamental division algorithm for polynomials.

Flashcard 73: What is $\frac{x^3-1}{x^2+x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x-1$. Factor $x^3-1=(x-1)(x^2+x+1)$ then cancel denominator.

Flashcard 74: What is $\frac{x^3+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

Answer: $x+\frac{-x+1}{x^2+1}$. Long division gives quotient $x$ with remainder $-x+1$.