Deconstructing Complicated Expressions - Algebra 2
Card 1 of 30
Which part of $Qigl(1+ rac{r}{n}igr)^{nt}$ is independent of $Q$?
Which part of $Qigl(1+ rac{r}{n}igr)^{nt}$ is independent of $Q$?
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$igl(1+ rac{r}{n}igr)^{nt}$. The compound interest factor doesn't depend on principal $Q$.
$igl(1+ rac{r}{n}igr)^{nt}$. The compound interest factor doesn't depend on principal $Q$.
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Which part of $2xigl(x^2+3x+1igr)$ is best viewed as one entity multiplied by $2x$?
Which part of $2xigl(x^2+3x+1igr)$ is best viewed as one entity multiplied by $2x$?
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$igl(x^2+3x+1igr)$. The polynomial is treated as one factor of $2x$.
$igl(x^2+3x+1igr)$. The polynomial is treated as one factor of $2x$.
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In $Aigl(1- rac{d}{100}igr)^k$, what factor represents repeated percent decrease?
In $Aigl(1- rac{d}{100}igr)^k$, what factor represents repeated percent decrease?
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$igl(1- rac{d}{100}igr)^k$. This represents the decay factor applied $k$ times.
$igl(1- rac{d}{100}igr)^k$. This represents the decay factor applied $k$ times.
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In $f(x)=-(x+4)^2+9$, what single entity is being squared?
In $f(x)=-(x+4)^2+9$, what single entity is being squared?
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$(x+4)$. The shifted variable is squared with negative coefficient.
$(x+4)$. The shifted variable is squared with negative coefficient.
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Identify $u$ so that $(x-4)^2-16$ can be viewed as $u^2-4^2$.
Identify $u$ so that $(x-4)^2-16$ can be viewed as $u^2-4^2$.
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$u=x-4$. Setting $u = x-4$ transforms this into $u^2 - 16$.
$u=x-4$. Setting $u = x-4$ transforms this into $u^2 - 16$.
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In $\frac{1}{2}(h+7)$, what single entity is scaled by $\frac{1}{2}$?
In $\frac{1}{2}(h+7)$, what single entity is scaled by $\frac{1}{2}$?
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$(h+7)$. The binomial is multiplied by the fractional coefficient.
$(h+7)$. The binomial is multiplied by the fractional coefficient.
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Which part of $P(1+r)^n$ can be viewed as a single factor independent of $P$?
Which part of $P(1+r)^n$ can be viewed as a single factor independent of $P$?
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$(1+r)^n$. This is the growth factor that doesn't contain $P$.
$(1+r)^n$. This is the growth factor that doesn't contain $P$.
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In $igl(2(x-3)igr)^2$, what single entity is being squared as a whole?
In $igl(2(x-3)igr)^2$, what single entity is being squared as a whole?
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$2(x-3)$. The entire product is treated as one unit being squared.
$2(x-3)$. The entire product is treated as one unit being squared.
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In $x^2-6x+9$, which single entity makes it recognizable as a perfect square?
In $x^2-6x+9$, which single entity makes it recognizable as a perfect square?
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Treat it as $(x-3)^2$ with entity $(x-3)$. Recognizing the perfect square trinomial pattern.
Treat it as $(x-3)^2$ with entity $(x-3)$. Recognizing the perfect square trinomial pattern.
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Identify $u$ so that $4x^2+12x+9$ can be viewed as $u^2$.
Identify $u$ so that $4x^2+12x+9$ can be viewed as $u^2$.
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$u=2x+3$. This makes the expression $(2x+3)^2$.
$u=2x+3$. This makes the expression $(2x+3)^2$.
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In $(x^2-1)(x^2+1)$, what single entity can replace $x^2$ to simplify viewing the product?
In $(x^2-1)(x^2+1)$, what single entity can replace $x^2$ to simplify viewing the product?
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Let $u=x^2$. Substitution simplifies the product structure.
Let $u=x^2$. Substitution simplifies the product structure.
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If $u=x^2$, how does $x^4-5x^2+6$ rewrite in terms of $u$?
If $u=x^2$, how does $x^4-5x^2+6$ rewrite in terms of $u$?
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$u^2-5u+6$. Substituting transforms this into a quadratic in $u$.
$u^2-5u+6$. Substituting transforms this into a quadratic in $u$.
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If $u=x+1$, how does $(x+1)^2+3(x+1)$ rewrite in terms of $u$?
If $u=x+1$, how does $(x+1)^2+3(x+1)$ rewrite in terms of $u$?
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$u^2+3u$. Direct substitution creates a simpler expression in $u$.
$u^2+3u$. Direct substitution creates a simpler expression in $u$.
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Identify $u$ so that $igl(3x-4igr)^2+2igl(3x-4igr)$ rewrites as $u^2+2u$.
Identify $u$ so that $igl(3x-4igr)^2+2igl(3x-4igr)$ rewrites as $u^2+2u$.
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$u=3x-4$. This substitution creates the standard quadratic form.
$u=3x-4$. This substitution creates the standard quadratic form.
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In $kigl( rac{1}{2}igr)^t$, what is the factor independent of $k$?
In $kigl( rac{1}{2}igr)^t$, what is the factor independent of $k$?
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$igl( rac{1}{2}igr)^t$. The exponential decay factor independent of the coefficient.
$igl( rac{1}{2}igr)^t$. The exponential decay factor independent of the coefficient.
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What does the expression $P(1+r)^n$ represent in words if $r$ is a growth rate?
What does the expression $P(1+r)^n$ represent in words if $r$ is a growth rate?
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Initial amount $P$ times growth factor $(1+r)^n$. Standard compound growth formula interpretation.
Initial amount $P$ times growth factor $(1+r)^n$. Standard compound growth formula interpretation.
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What does the expression $P(1-r)^n$ represent in words if $0<r<1$?
What does the expression $P(1-r)^n$ represent in words if $0<r<1$?
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Initial amount $P$ times decay factor $(1-r)^n$. Standard exponential decay formula interpretation.
Initial amount $P$ times decay factor $(1-r)^n$. Standard exponential decay formula interpretation.
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Identify the single entity in $3igl(2x^2-5x+1igr)$ that is multiplied by $3$.
Identify the single entity in $3igl(2x^2-5x+1igr)$ that is multiplied by $3$.
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$2x^2-5x+1$. The polynomial expression is scaled by the coefficient $3$.
$2x^2-5x+1$. The polynomial expression is scaled by the coefficient $3$.
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Identify the single entity in $ rac{(x-1)^2}{9}$ that is being divided by $9$.
Identify the single entity in $ rac{(x-1)^2}{9}$ that is being divided by $9$.
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$(x-1)^2$. The squared binomial is the numerator being divided.
$(x-1)^2$. The squared binomial is the numerator being divided.
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Which subexpression acts as one unit in $igl( rac{x+1}{x-2}igr)^3$?
Which subexpression acts as one unit in $igl( rac{x+1}{x-2}igr)^3$?
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$ rac{x+1}{x-2}$. The rational expression is raised to the third power.
$ rac{x+1}{x-2}$. The rational expression is raised to the third power.
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In $a(bc+d)$, which part is best viewed as a single entity multiplied by $a$?
In $a(bc+d)$, which part is best viewed as a single entity multiplied by $a$?
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$(bc+d)$. The sum is treated as a single factor of $a$.
$(bc+d)$. The sum is treated as a single factor of $a$.
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Identify the single entity in $(m+n)(m-n)$ that suggests a product of two units.
Identify the single entity in $(m+n)(m-n)$ that suggests a product of two units.
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$(m+n)$ and $(m-n)$. Two separate binomial entities multiplied together.
$(m+n)$ and $(m-n)$. Two separate binomial entities multiplied together.
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In $(x^2+1)^2-4$, what subexpression is most natural to treat as one unit?
In $(x^2+1)^2-4$, what subexpression is most natural to treat as one unit?
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$(x^2+1)$. The quadratic expression is the base of operations.
$(x^2+1)$. The quadratic expression is the base of operations.
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What single entity makes $u^2-9$ recognizable as a difference of squares?
What single entity makes $u^2-9$ recognizable as a difference of squares?
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Treat $u$ as one unit: $u^2-3^2$. Substituting $u$ reveals the difference of squares pattern.
Treat $u$ as one unit: $u^2-3^2$. Substituting $u$ reveals the difference of squares pattern.
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Identify $u$ so that $(x-4)^2-16$ can be viewed as $u^2-4^2$.
Identify $u$ so that $(x-4)^2-16$ can be viewed as $u^2-4^2$.
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$u=x-4$. Setting $u = x-4$ transforms this into $u^2 - 16$.
$u=x-4$. Setting $u = x-4$ transforms this into $u^2 - 16$.
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Identify $u$ so that $(2x+1)^2-25$ can be viewed as $u^2-5^2$.
Identify $u$ so that $(2x+1)^2-25$ can be viewed as $u^2-5^2$.
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$u=2x+1$. This substitution reveals the difference of squares structure.
$u=2x+1$. This substitution reveals the difference of squares structure.
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Identify $u$ so that $(x^2+3)^2-1$ can be viewed as $u^2-1^2$.
Identify $u$ so that $(x^2+3)^2-1$ can be viewed as $u^2-1^2$.
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$u=x^2+3$. The quadratic plus constant becomes the single entity.
$u=x^2+3$. The quadratic plus constant becomes the single entity.
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In $(x+2)^3+(x+2)^2$, what common single entity should be factored out?
In $(x+2)^3+(x+2)^2$, what common single entity should be factored out?
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$(x+2)^2$. The squared binomial can be factored from both terms.
$(x+2)^2$. The squared binomial can be factored from both terms.
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In $5(3t-1)+2(3t-1)$, what single entity is common to both terms?
In $5(3t-1)+2(3t-1)$, what single entity is common to both terms?
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$(3t-1)$. The binomial appears in both terms as a common factor.
$(3t-1)$. The binomial appears in both terms as a common factor.
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What is the shared single entity in $a(x-7)-b(x-7)$?
What is the shared single entity in $a(x-7)-b(x-7)$?
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$(x-7)$. The binomial is multiplied by different coefficients.
$(x-7)$. The binomial is multiplied by different coefficients.
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