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 $\frac{x^3+x^2}{x^2+1}$ 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 2: 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 3: 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 4: What is the remainder when $a(x)$ is divided by $x+3$?

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

Flashcard 5: What is the remainder when $a(x)$ is divided by $x$?

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

Flashcard 6: 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 7: 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 8: 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 9: 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 10: 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 11: 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 12: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: What is the remainder when $a(x)$ is divided by $x$?

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

Flashcard 19: 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 20: 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 21: What is a quick method (instead of long division) for dividing by $x-c$?

Answer: Synthetic division. Efficient method for linear divisors.

Flashcard 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: 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.