Algebra 2 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 2.

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 $\frac{x^3-1}{x-1}$ rewritten as $q(x)+\frac{r(x)}{x-1}$?

Answer: $x^2+x+1+\frac{0}{x-1}$. Use identity $x^3-1=(x-1)(x^2+x+1)$ for exact division.

Flashcard 2: What is $\frac{6x^2+7x-3}{3x+1}$ rewritten as $q(x)+\frac{r(x)}{3x+1}$?

Answer: $2x+\frac{5x-3}{3x+1}$. Divide quadratic by linear expression $3x+1$.

Flashcard 3: What identity rewrites $a(x)$ after division by $b(x)$?

Answer: $a(x)=b(x)q(x)+r(x)$. Division algorithm identity relating dividend, divisor, quotient, and remainder.

Flashcard 4: What must you do if $a(x)$ is missing an $x^2$ term before long division?

Answer: Insert $0x^2$ as a placeholder. Missing terms need zero coefficients for proper alignment.

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

Answer: $x^2-2x+1+\frac{0}{x-1}$. Perfect cube $(x-1)^3$ divided by $(x-1)$.

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

Answer: $x+1+\frac{-x-1}{x^2+1}$. Factor $x^2$ and divide by $x^2+1$.

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

Answer: $x+1+\frac{0}{x-1}$. Factor $x^2-1=(x-1)(x+1)$ for exact division.

Flashcard 8: What is $\frac{x^2-1}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?

Answer: $x-1+\frac{0}{x+1}$. Factor $x^2-1=(x+1)(x-1)$ for exact division.

Flashcard 9: What is $\frac{x^3}{x^2+1}$ rewritten as $q(x)+\frac{r(x)}{x^2+1}$?

Answer: $x+\frac{-x}{x^2+1}$. Divide cubic by quadratic to get linear quotient.

Flashcard 10: Identify the correct rewrite: $\frac{a(x)}{b(x)}$ equals what expression using $q(x)$ and $r(x)$?

Answer: $q(x)+\frac{r(x)}{b(x)}$. Standard polynomial division form.

Flashcard 11: What condition must $r(x)$ satisfy 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 12: What is $\frac{3x^2-2x+4}{x-1}$ rewritten as $q(x)+\frac{r(x)}{x-1}$?

Answer: $3x+1+\frac{5}{x-1}$. Polynomial division with quotient degree 1.

Flashcard 13: What is $\frac{x^3+2x^2+x+1}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?

Answer: $x^2+x+\frac{1}{x+1}$. Long division of cubic by linear expression.

Flashcard 14: What is the remainder theorem statement for dividing by $x-c$?

Answer: Remainder is $a(c)$. Evaluating the polynomial at $c$ gives the remainder.

Flashcard 15: What form results from dividing polynomials: $\frac{a(x)}{b(x)}= ?$

Answer: $q(x)+\frac{r(x)}{b(x)}$. Quotient plus remainder over divisor form.

Flashcard 16: What is a quick method (instead of long division) for dividing by $x-c$?

Answer: Synthetic division. Efficient method for linear divisors.

Flashcard 17: What is $\frac{x^2+2x+5}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?

Answer: $x+1+\frac{4}{x+1}$. Long division yields quotient $x+1$ and remainder $4$.

Flashcard 18: What is $\frac{x^4+2x^3+3}{x^2}$ rewritten as $q(x)+\frac{r(x)}{x^2}$?

Answer: $x^2+2x+\frac{3}{x^2}$. Divide each term of the numerator by $x^2$.

Flashcard 19: What is $\frac{x^3-x^2+2}{x-1}$ rewritten as $q(x)+\frac{r(x)}{x-1}$?

Answer: $x^2+\frac{2}{x-1}$. Divide cubic by linear to get quadratic quotient.

Flashcard 20: What must be true about $a(x)$ and $b(x)$ to start long division of $\frac{a(x)}{b(x)}$?

Answer: Write both in descending powers, include $0$ terms. Standard form needed for the division algorithm.

Flashcard 21: What is $\frac{x^3-8}{x-2}$ rewritten as $q(x)+\frac{r(x)}{x-2}$?

Answer: $x^2+2x+4+\frac{0}{x-2}$. Use identity $x^3-8=(x-2)(x^2+2x+4)$ for exact division.

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

Answer: $x^2-x+1+\frac{0}{x+1}$. Use identity $x^3+1=(x+1)(x^2-x+1)$ for exact division.

Flashcard 23: What form results from dividing polynomials: $\frac{a(x)}{b(x)}=\ ?$

Answer: $q(x)+\frac{r(x)}{b(x)}$. Quotient plus remainder over divisor form.

Flashcard 24: What is the name of the process used to write $\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$?

Answer: Polynomial long division. Standard algorithm for dividing polynomials.

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

Answer: $x+2+\frac{-2x+1}{x^2+1}$. Divide cubic by quadratic using long division.

Flashcard 26: What is $\frac{x^3+2x^2+4x+8}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?

Answer: $x^2+4+\frac{0}{x+2}$. Factor by grouping: $(x+2)(x^2+4)$ for exact division.

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

Answer: $x+2+\frac{0}{x-2}$. Factor $x^2-4=(x-2)(x+2)$ for exact division.

Flashcard 28: What is $\frac{x^3+4}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?

Answer: $x^2-x+1+\frac{3}{x+1}$. Divide cubic by linear using long division.

Flashcard 29: What is $\frac{4x^2+1}{2x-1}$ rewritten as $q(x)+\frac{r(x)}{2x-1}$?

Answer: $2x+1+\frac{2}{2x-1}$. Divide by the linear expression $2x-1$.

Flashcard 30: What is $\frac{2x^2+3x-7}{x-2}$ rewritten as $q(x)+\frac{r(x)}{x-2}$?

Answer: $2x+7+\frac{7}{x-2}$. Polynomial division with quadratic quotient.

Flashcard 31: What is $r(x)$ when $b(x)$ divides $a(x)$ evenly in $\frac{a(x)}{b(x)}$?

Answer: $r(x)=0$. No remainder when division is exact.

Flashcard 32: What is the quotient called when dividing by $x-c$ using synthetic division?

Answer: The depressed polynomial $q(x)$. The quotient after factoring out $(x-c)$.

Flashcard 33: What is $\frac{x^2+4x+7}{x-1}$ rewritten as $q(x)+\frac{r(x)}{x-1}$?

Answer: $x+5+\frac{12}{x-1}$. Long division with quotient $x+5$ and remainder $12$.

Flashcard 34: What is $\frac{x^3-4x}{x^2-1}$ rewritten as $q(x)+\frac{r(x)}{x^2-1}$?

Answer: $x+\frac{-3x}{x^2-1}$. Factor out $x$ first: $x^3-4x=x(x^2-4)$.

Flashcard 35: What is the first step to find the leading term of $q(x)$ in long division?

Answer: Divide leading term of $a(x)$ by leading term of $b(x)$. Divide highest degree terms first.

Flashcard 36: What is the degree of the quotient when dividing degree $n$ by degree $m$ with $n\ge m$?

Answer: $\deg(q(x))=n-m$. Quotient degree equals dividend minus divisor degree.

Flashcard 37: What is $\frac{x^2-4x+7}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?

Answer: $x-5+\frac{12}{x+1}$. Divide quadratic by $x+1$ using polynomial division.

Flashcard 38: What is $\frac{x^2+1}{x}$ rewritten as $q(x)+\frac{r(x)}{x}$?

Answer: $x+\frac{1}{x}$. Divide to get quotient $x$ and remainder $1$.

Flashcard 39: What is $q(x)$ when $\deg(a(x))<\deg(b(x))$ for $\frac{a(x)}{b(x)}$?

Answer: $q(x)=0$. No quotient when dividend degree is smaller.

Flashcard 40: What is $\frac{x^2+3}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?

Answer: $x-1+\frac{4}{x+1}$. Divide $x^2+3$ by $x+1$ using polynomial long division.

Flashcard 41: What is $\frac{x^2-4}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?

Answer: $x-2+\frac{0}{x+2}$. Factor $x^2-4=(x+2)(x-2)$ for exact division.

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

Answer: $x^2+\frac{1}{x}$. Simplify by dividing each term by $x^2$.

Flashcard 43: What is $\frac{x^3-2x^2+4x-8}{x-2}$ rewritten as $q(x)+\frac{r(x)}{x-2}$?

Answer: $x^2+4+\frac{0}{x-2}$. Factor by grouping: $(x-2)(x^2+4)$ for exact division.

Flashcard 44: What is $\frac{x^2+6x}{x+3}$ rewritten as $q(x)+\frac{r(x)}{x+3}$?

Answer: $x+3+\frac{-9}{x+3}$. Rewrite as $\frac{x(x+6)}{x+3}$ and divide.

Flashcard 45: What is $\frac{x^3-2x+1}{x^2+x}$ rewritten as $q(x)+\frac{r(x)}{x^2+x}$?

Answer: $x-1+\frac{-x+1}{x^2+x}$. Divide cubic by quadratic $x^2+x=x(x+1)$.

Flashcard 46: What is $\frac{x^2+1}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?

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

Flashcard 47: What is $\frac{x^2+3x+2}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?

Answer: $x+2+\frac{0}{x+1}$. Factor $x^2+3x+2=(x+1)(x+2)$ for exact division.

Flashcard 48: What is $\frac{2x^2+5x+1}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?

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

Flashcard 49: What is $\frac{2x^3+3x^2-x+5}{x^2-1}$ rewritten as $q(x)+\frac{r(x)}{x^2-1}$?

Answer: $2x+3+\frac{x+8}{x^2-1}$. Long division of cubic by quadratic expression.

Flashcard 50: What is $\frac{x^3+8}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?

Answer: $x^2-2x+4+\frac{0}{x+2}$. Use identity $x^3+8=(x+2)(x^2-2x+4)$ for exact division.

Flashcard 51: What is the remainder when $a(x)$ is divided by $x+3$?

Answer: $a(-3)$. Apply remainder theorem with $x+3=(x-(-3))$.

Flashcard 52: What is the remainder when $a(x)$ is divided by $x$?

Answer: $a(0)$. Substitute $x=0$ using the remainder theorem.

Flashcard 53: What is $\frac{5x^2+10x+3}{5x+5}$ rewritten as $q(x)+\frac{r(x)}{5x+5}$?

Answer: $x+1+\frac{-2}{5x+5}$. Factor out 5 and divide: $\frac{5(x^2+2x)+3}{5(x+1)}$.

Flashcard 54: What is $\frac{x^3+4x^2+5x+2}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?

Answer: $x^2+2x+1+\frac{0}{x+2}$. Exact division since $(x+2)$ is a factor.

Flashcard 55: What is $\frac{x^4-1}{x^2+1}$ rewritten as $q(x)+\frac{r(x)}{x^2+1}$?

Answer: $x^2-1+\frac{0}{x^2+1}$. Use identity $x^4-1=(x^2+1)(x^2-1)$ for exact division.

Flashcard 56: What is $\frac{x^2}{x-3}$ rewritten as $q(x)+\frac{r(x)}{x-3}$?

Answer: $x+3+\frac{9}{x-3}$. Add and subtract 9: $x^2=(x-3)(x+3)+9$.