Rewriting Rational Expressions - Algebra
Card 1 of 30
What is $\frac{x^3+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^3+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x+\frac{-x+1}{x^2+1}$. Long division gives quotient $x$ with remainder $-x+1$.
$x+\frac{-x+1}{x^2+1}$. Long division gives quotient $x$ with remainder $-x+1$.
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What is $\frac{x^3+1}{x^2-x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^3+1}{x^2-x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x+1$. Factor $x^3+1=(x+1)(x^2-x+1)$ then cancel denominator.
$x+1$. Factor $x^3+1=(x+1)(x^2-x+1)$ then cancel denominator.
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What is the name of the process used to rewrite $\frac{a(x)}{b(x)}$ as $q(x)+\frac{r(x)}{b(x)}$?
What is the name of the process used to rewrite $\frac{a(x)}{b(x)}$ as $q(x)+\frac{r(x)}{b(x)}$?
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Polynomial long division. The systematic method for dividing polynomials with remainder.
Polynomial long division. The systematic method for dividing polynomials with remainder.
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What is $q(x)$ called in $a(x)=b(x)q(x)+r(x)$?
What is $q(x)$ called in $a(x)=b(x)q(x)+r(x)$?
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The quotient polynomial $q(x)$. The result when $a(x)$ is divided by $b(x)$.
The quotient polynomial $q(x)$. The result when $a(x)$ is divided by $b(x)$.
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What is $r(x)$ called in $a(x)=b(x)q(x)+r(x)$?
What is $r(x)$ called in $a(x)=b(x)q(x)+r(x)$?
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The remainder polynomial $r(x)$. What's left after division when $\deg(r)<\deg(b)$.
The remainder polynomial $r(x)$. What's left after division when $\deg(r)<\deg(b)$.
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What is the identity that connects division form and remainder form for polynomials?
What is the identity that connects division form and remainder form for polynomials?
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$a(x)=b(x)q(x)+r(x)$. The fundamental division algorithm for polynomials.
$a(x)=b(x)q(x)+r(x)$. The fundamental division algorithm for polynomials.
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What is the rewritten form of $\frac{a(x)}{b(x)}$ after division with remainder?
What is the rewritten form of $\frac{a(x)}{b(x)}$ after division with remainder?
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$\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$. Standard quotient-plus-remainder form after polynomial division.
$\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$. Standard quotient-plus-remainder form after polynomial division.
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What must you do first before long dividing by $x^2+3$ if terms are missing in $a(x)$?
What must you do first before long dividing by $x^2+3$ if terms are missing in $a(x)$?
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Insert missing powers with coefficient $0$. Ensures all terms are present for proper column alignment.
Insert missing powers with coefficient $0$. Ensures all terms are present for proper column alignment.
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What is the remainder degree condition when dividing by a linear $b(x)$ with $\deg(b(x))=1$?
What is the remainder degree condition when dividing by a linear $b(x)$ with $\deg(b(x))=1$?
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$r(x)$ is a constant (degree $0$). Linear divisor means remainder has degree less than 1.
$r(x)$ is a constant (degree $0$). Linear divisor means remainder has degree less than 1.
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What is the remainder degree condition when dividing by a quadratic $b(x)$ with $\deg(b(x))=2$?
What is the remainder degree condition when dividing by a quadratic $b(x)$ with $\deg(b(x))=2$?
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$r(x)$ has form $ax+b$. Quadratic divisor means remainder has degree less than 2.
$r(x)$ has form $ax+b$. Quadratic divisor means remainder has degree less than 2.
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What is the key stopping condition in polynomial long division?
What is the key stopping condition in polynomial long division?
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Stop when $\deg(\text{remainder})<\deg(b(x))$. Division ends when remainder degree drops below divisor degree.
Stop when $\deg(\text{remainder})<\deg(b(x))$. Division ends when remainder degree drops below divisor degree.
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What is $\frac{x^2+1}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^2+1}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x+\frac{1}{x}$. Split each term: $\frac{x^2}{x}+\frac{1}{x}=x+\frac{1}{x}$.
$x+\frac{1}{x}$. Split each term: $\frac{x^2}{x}+\frac{1}{x}=x+\frac{1}{x}$.
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What is $\frac{x^2+3x+2}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^2+3x+2}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x+3+\frac{2}{x}$. Split each term: $\frac{x^2}{x}+\frac{3x}{x}+\frac{2}{x}$.
$x+3+\frac{2}{x}$. Split each term: $\frac{x^2}{x}+\frac{3x}{x}+\frac{2}{x}$.
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What is $\frac{3x^2+6}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{3x^2+6}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$3x+\frac{6}{x}$. Split each term: $\frac{3x^2}{x}+\frac{6}{x}=3x+\frac{6}{x}$.
$3x+\frac{6}{x}$. Split each term: $\frac{3x^2}{x}+\frac{6}{x}=3x+\frac{6}{x}$.
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What is $\frac{x^3+2x}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^3+2x}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x^2+2$. Split each term: $\frac{x^3}{x}+\frac{2x}{x}=x^2+2$.
$x^2+2$. Split each term: $\frac{x^3}{x}+\frac{2x}{x}=x^2+2$.
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What is $\frac{5x^3-10x}{5x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{5x^3-10x}{5x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x^2-2$. Split each term: $\frac{5x^3}{5x}-\frac{10x}{5x}=x^2-2$.
$x^2-2$. Split each term: $\frac{5x^3}{5x}-\frac{10x}{5x}=x^2-2$.
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What is $\frac{x^2-4}{x-2}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^2-4}{x-2}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x+2$. Factor as $(x-2)(x+2)$ then cancel $(x-2)$.
$x+2$. Factor as $(x-2)(x+2)$ then cancel $(x-2)$.
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What is $\frac{x^2+2x+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^2+2x+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x+1$. Factor as $(x+1)^2$ then cancel $(x+1)$.
$x+1$. Factor as $(x+1)^2$ then cancel $(x+1)$.
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What is $\frac{x^2+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^2+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x-1+\frac{2}{x+1}$. Long division gives quotient $x-1$ with remainder $2$.
$x-1+\frac{2}{x+1}$. Long division gives quotient $x-1$ with remainder $2$.
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What is $\frac{x^2+5x+6}{x+2}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^2+5x+6}{x+2}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x+3$. Factor as $(x+2)(x+3)$ then cancel $(x+2)$.
$x+3$. Factor as $(x+2)(x+3)$ then cancel $(x+2)$.
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What is $\frac{x^2+5x+5}{x+2}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^2+5x+5}{x+2}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x+3-\frac{1}{x+2}$. Long division gives quotient $x+3$ with remainder $-1$.
$x+3-\frac{1}{x+2}$. Long division gives quotient $x+3$ with remainder $-1$.
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What is $\frac{2x^2+3x+4}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{2x^2+3x+4}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$2x+1+\frac{3}{x+1}$. Long division gives quotient $2x+1$ with remainder $3$.
$2x+1+\frac{3}{x+1}$. Long division gives quotient $2x+1$ with remainder $3$.
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What is $\frac{x^4-x}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^4-x}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x^3-1$. Factor out $x$: $\frac{x(x^3-1)}{x}=x^3-1$.
$x^3-1$. Factor out $x$: $\frac{x(x^3-1)}{x}=x^3-1$.
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What is $\frac{x^2-1}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^2-1}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x+1$. Factor as $(x-1)(x+1)$ then cancel $(x-1)$.
$x+1$. Factor as $(x-1)(x+1)$ then cancel $(x-1)$.
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What is $\frac{x^3-1}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^3-1}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x^2+x+1$. Factor $x^3-1=(x-1)(x^2+x+1)$ then cancel $(x-1)$.
$x^2+x+1$. Factor $x^3-1=(x-1)(x^2+x+1)$ then cancel $(x-1)$.
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What is $\frac{x^3+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^3+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
Tap to reveal answer
$x^2-x+1$. Factor $x^3+1=(x+1)(x^2-x+1)$ then cancel $(x+1)$.
$x^2-x+1$. Factor $x^3+1=(x+1)(x^2-x+1)$ then cancel $(x+1)$.
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What is $\frac{x^3+2}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^3+2}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x^2+\frac{2}{x}$. Split each term: $\frac{x^3}{x}+\frac{2}{x}=x^2+\frac{2}{x}$.
$x^2+\frac{2}{x}$. Split each term: $\frac{x^3}{x}+\frac{2}{x}=x^2+\frac{2}{x}$.
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What is $\frac{x^3+2x^2+3}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^3+2x^2+3}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x+2+\frac{3}{x^2}$. Split each term: $\frac{x^3}{x^2}+\frac{2x^2}{x^2}+\frac{3}{x^2}$.
$x+2+\frac{3}{x^2}$. Split each term: $\frac{x^3}{x^2}+\frac{2x^2}{x^2}+\frac{3}{x^2}$.
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What is $\frac{x^3+2x+1}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^3+2x+1}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?
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$x+\frac{2x+1}{x^2}$. Split first term then combine remainder: $x+\frac{2x+1}{x^2}$.
$x+\frac{2x+1}{x^2}$. Split first term then combine remainder: $x+\frac{2x+1}{x^2}$.
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What is $\frac{x^3+x^2+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
What is $\frac{x^3+x^2+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?
Tap to reveal answer
$x+\frac{x^2-x+1}{x^2+1}$. Long division gives quotient $x$ with remainder $x^2-x+1$.
$x+\frac{x^2-x+1}{x^2+1}$. Long division gives quotient $x$ with remainder $x^2-x+1$.
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