Rewriting Rational Expressions - Algebra 2
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What is $\frac{x^3+x^2}{x^2+1}$ rewritten as $q(x)+\frac{r(x)}{x^2+1}$?
What is $\frac{x^3+x^2}{x^2+1}$ rewritten as $q(x)+\frac{r(x)}{x^2+1}$?
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$x+1+\frac{-x-1}{x^2+1}$. Factor $x^2$ and divide by $x^2+1$.
$x+1+\frac{-x-1}{x^2+1}$. Factor $x^2$ and divide by $x^2+1$.
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What is $\frac{x^3}{x^2+1}$ rewritten as $q(x)+\frac{r(x)}{x^2+1}$?
What is $\frac{x^3}{x^2+1}$ rewritten as $q(x)+\frac{r(x)}{x^2+1}$?
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$x+\frac{-x}{x^2+1}$. Divide cubic by quadratic to get linear quotient.
$x+\frac{-x}{x^2+1}$. Divide cubic by quadratic to get linear quotient.
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What must be true about $a(x)$ and $b(x)$ to start long division of $\frac{a(x)}{b(x)}$?
What must be true about $a(x)$ and $b(x)$ to start long division of $\frac{a(x)}{b(x)}$?
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Write both in descending powers, include $0$ terms. Standard form needed for the division algorithm.
Write both in descending powers, include $0$ terms. Standard form needed for the division algorithm.
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What is the remainder when $a(x)$ is divided by $x+3$?
What is the remainder when $a(x)$ is divided by $x+3$?
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$a(-3)$. Apply remainder theorem with $x+3=(x-(-3))$.
$a(-3)$. Apply remainder theorem with $x+3=(x-(-3))$.
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What is the remainder when $a(x)$ is divided by $x$?
What is the remainder when $a(x)$ is divided by $x$?
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$a(0)$. Substitute $x=0$ using the remainder theorem.
$a(0)$. Substitute $x=0$ using the remainder theorem.
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What is $r(x)$ when $b(x)$ divides $a(x)$ evenly in $\frac{a(x)}{b(x)}$?
What is $r(x)$ when $b(x)$ divides $a(x)$ evenly in $\frac{a(x)}{b(x)}$?
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$r(x)=0$. No remainder when division is exact.
$r(x)=0$. No remainder when division is exact.
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What is $\frac{x^3-3x^2+3x-1}{x-1}$ rewritten as $q(x)+\frac{r(x)}{x-1}$?
What is $\frac{x^3-3x^2+3x-1}{x-1}$ rewritten as $q(x)+\frac{r(x)}{x-1}$?
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$x^2-2x+1+\frac{0}{x-1}$. Perfect cube $ (x-1)^3 $ divided by $ (x-1) $.
$x^2-2x+1+\frac{0}{x-1}$. Perfect cube $ (x-1)^3 $ divided by $ (x-1) $.
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What is $\frac{x^3+2x^2+4x+8}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?
What is $\frac{x^3+2x^2+4x+8}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?
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$x^2+4+\frac{0}{x+2}$. Factor by grouping: $(x+2)(x^2+4)$ for exact division.
$x^2+4+\frac{0}{x+2}$. Factor by grouping: $(x+2)(x^2+4)$ for exact division.
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What is the name of the process used to write $\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$?
What is the name of the process used to write $\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$?
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Polynomial long division. Standard algorithm for dividing polynomials.
Polynomial long division. Standard algorithm for dividing polynomials.
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What form results from dividing polynomials: $\frac{a(x)}{b(x)}=\ ?$
What form results from dividing polynomials: $\frac{a(x)}{b(x)}=\ ?$
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$q(x)+\frac{r(x)}{b(x)}$. Quotient plus remainder over divisor form.
$q(x)+\frac{r(x)}{b(x)}$. Quotient plus remainder over divisor form.
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What must be true about $a(x)$ and $b(x)$ to start long division of $\frac{a(x)}{b(x)}$?
What must be true about $a(x)$ and $b(x)$ to start long division of $\frac{a(x)}{b(x)}$?
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Write both in descending powers, include $0$ terms. Standard form needed for the division algorithm.
Write both in descending powers, include $0$ terms. Standard form needed for the division algorithm.
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What is $q(x)$ when $\deg(a(x))<\deg(b(x))$ for $\frac{a(x)}{b(x)}$?
What is $q(x)$ when $\deg(a(x))<\deg(b(x))$ for $\frac{a(x)}{b(x)}$?
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$q(x)=0$. No quotient when dividend degree is smaller.
$q(x)=0$. No quotient when dividend degree is smaller.
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What is the degree of the quotient when dividing degree $n$ by degree $m$ with $n\ge m$?
What is the degree of the quotient when dividing degree $n$ by degree $m$ with $n\ge m$?
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$\deg(q(x))=n-m$. Quotient degree equals dividend minus divisor degree.
$\deg(q(x))=n-m$. Quotient degree equals dividend minus divisor degree.
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What is the first step to find the leading term of $q(x)$ in long division?
What is the first step to find the leading term of $q(x)$ in long division?
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Divide leading term of $a(x)$ by leading term of $b(x)$. Divide highest degree terms first.
Divide leading term of $a(x)$ by leading term of $b(x)$. Divide highest degree terms first.
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What identity rewrites $a(x)$ after division by $b(x)$?
What identity rewrites $a(x)$ after division by $b(x)$?
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$a(x)=b(x)q(x)+r(x)$. Division algorithm identity relating dividend, divisor, quotient, and remainder.
$a(x)=b(x)q(x)+r(x)$. Division algorithm identity relating dividend, divisor, quotient, and remainder.
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What is the remainder theorem statement for dividing by $x-c$?
What is the remainder theorem statement for dividing by $x-c$?
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Remainder is $a(c)$. Evaluating the polynomial at $c$ gives the remainder.
Remainder is $a(c)$. Evaluating the polynomial at $c$ gives the remainder.
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What is the quotient called when dividing by $x-c$ using synthetic division?
What is the quotient called when dividing by $x-c$ using synthetic division?
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The depressed polynomial $q(x)$. The quotient after factoring out $(x-c)$.
The depressed polynomial $q(x)$. The quotient after factoring out $(x-c)$.
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What is the remainder when $a(x)$ is divided by $x$?
What is the remainder when $a(x)$ is divided by $x$?
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$a(0)$. Substitute $x=0$ using the remainder theorem.
$a(0)$. Substitute $x=0$ using the remainder theorem.
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Identify the correct rewrite: $\frac{a(x)}{b(x)}$ equals what expression using $q(x)$ and $r(x)$?
Identify the correct rewrite: $\frac{a(x)}{b(x)}$ equals what expression using $q(x)$ and $r(x)$?
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$q(x)+\frac{r(x)}{b(x)}$. Standard polynomial division form.
$q(x)+\frac{r(x)}{b(x)}$. Standard polynomial division form.
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What must you do if $a(x)$ is missing an $x^2$ term before long division?
What must you do if $a(x)$ is missing an $x^2$ term before long division?
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Insert $0x^2$ as a placeholder. Missing terms need zero coefficients for proper alignment.
Insert $0x^2$ as a placeholder. Missing terms need zero coefficients for proper alignment.
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What is a quick method (instead of long division) for dividing by $x-c$?
What is a quick method (instead of long division) for dividing by $x-c$?
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Synthetic division. Efficient method for linear divisors.
Synthetic division. Efficient method for linear divisors.
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What is $\frac{x^2+4x+7}{x-1}$ rewritten as $q(x)+\frac{r(x)}{x-1}$?
What is $\frac{x^2+4x+7}{x-1}$ rewritten as $q(x)+\frac{r(x)}{x-1}$?
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$x+5+\frac{12}{x-1}$. Long division with quotient $x+5$ and remainder $12$.
$x+5+\frac{12}{x-1}$. Long division with quotient $x+5$ and remainder $12$.
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What is $\frac{x^2-4x+7}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?
What is $\frac{x^2-4x+7}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?
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$x-5+\frac{12}{x+1}$. Divide quadratic by $x+1$ using polynomial division.
$x-5+\frac{12}{x+1}$. Divide quadratic by $x+1$ using polynomial division.
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What is $\frac{x^3-2x^2+4x-8}{x-2}$ rewritten as $q(x)+\frac{r(x)}{x-2}$?
What is $\frac{x^3-2x^2+4x-8}{x-2}$ rewritten as $q(x)+\frac{r(x)}{x-2}$?
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$x^2+4+\frac{0}{x-2}$. Factor by grouping: $(x-2)(x^2+4)$ for exact division.
$x^2+4+\frac{0}{x-2}$. Factor by grouping: $(x-2)(x^2+4)$ for exact division.
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What condition must $r(x)$ satisfy in $\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$?
What condition must $r(x)$ satisfy in $\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$?
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$\deg(r(x))<\deg(b(x))$. The remainder degree must be less than the divisor degree.
$\deg(r(x))<\deg(b(x))$. The remainder degree must be less than the divisor degree.
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What is $\frac{x^4+2x^3+3}{x^2}$ rewritten as $q(x)+\frac{r(x)}{x^2}$?
What is $\frac{x^4+2x^3+3}{x^2}$ rewritten as $q(x)+\frac{r(x)}{x^2}$?
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$x^2+2x+\frac{3}{x^2}$. Divide each term of the numerator by $x^2$.
$x^2+2x+\frac{3}{x^2}$. Divide each term of the numerator by $x^2$.
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What is $\frac{6x^2+7x-3}{3x+1}$ rewritten as $q(x)+\frac{r(x)}{3x+1}$?
What is $\frac{6x^2+7x-3}{3x+1}$ rewritten as $q(x)+\frac{r(x)}{3x+1}$?
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$2x+\frac{5x-3}{3x+1}$. Divide quadratic by linear expression $3x+1$.
$2x+\frac{5x-3}{3x+1}$. Divide quadratic by linear expression $3x+1$.
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What is $\frac{x^3+8}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?
What is $\frac{x^3+8}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?
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$x^2-2x+4+\frac{0}{x+2}$. Use identity $x^3+8=(x+2)(x^2-2x+4)$ for exact division.
$x^2-2x+4+\frac{0}{x+2}$. Use identity $x^3+8=(x+2)(x^2-2x+4)$ for exact division.
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What is $\frac{x^3-8}{x-2}$ rewritten as $q(x)+\frac{r(x)}{x-2}$?
What is $\frac{x^3-8}{x-2}$ rewritten as $q(x)+\frac{r(x)}{x-2}$?
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$x^2+2x+4+\frac{0}{x-2}$. Use identity $x^3-8=(x-2)(x^2+2x+4)$ for exact division.
$x^2+2x+4+\frac{0}{x-2}$. Use identity $x^3-8=(x-2)(x^2+2x+4)$ for exact division.
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What is $\frac{x^2-1}{x-1}$ rewritten as $q(x)+\frac{r(x)}{x-1}$?
What is $\frac{x^2-1}{x-1}$ rewritten as $q(x)+\frac{r(x)}{x-1}$?
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$x+1+\frac{0}{x-1}$. Factor $x^2-1=(x-1)(x+1)$ for exact division.
$x+1+\frac{0}{x-1}$. Factor $x^2-1=(x-1)(x+1)$ for exact division.
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