Deconstructing Complicated Expressions - Algebra
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What is the factor that does not depend on $t$ in $t^2(4-3t)^5$?
What is the factor that does not depend on $t$ in $t^2(4-3t)^5$?
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None; both $t^2$ and $(4-3t)^5$ depend on $t$. Every factor contains the variable $t$, so no factor is independent.
None; both $t^2$ and $(4-3t)^5$ depend on $t$. Every factor contains the variable $t$, so no factor is independent.
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Which part is naturally treated as one entity in $5x(x^2+4x+1)$?
Which part is naturally treated as one entity in $5x(x^2+4x+1)$?
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$(x^2+4x+1)$. The trinomial in parentheses is treated as a single multiplicand.
$(x^2+4x+1)$. The trinomial in parentheses is treated as a single multiplicand.
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What is the repeated factor form of $(a+b)^3$ when treating $(a+b)$ as one entity?
What is the repeated factor form of $(a+b)^3$ when treating $(a+b)$ as one entity?
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$(a+b)(a+b)(a+b)$. The exponent indicates how many times $(a+b)$ multiplies itself.
$(a+b)(a+b)(a+b)$. The exponent indicates how many times $(a+b)$ multiplies itself.
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What is the constant factor (not depending on $x$) in $9(x+1)^2$?
What is the constant factor (not depending on $x$) in $9(x+1)^2$?
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$9$. The constant factor multiplies the entire expression $(x+1)^2$.
$9$. The constant factor multiplies the entire expression $(x+1)^2$.
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Identify the single entity that is being raised to a power in $(2x-5)^4$.
Identify the single entity that is being raised to a power in $(2x-5)^4$.
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$(2x-5)$. The expression inside the parentheses forms the base of the power.
$(2x-5)$. The expression inside the parentheses forms the base of the power.
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What is the coefficient of the entity $(x-3)$ in $7(x-3)$?
What is the coefficient of the entity $(x-3)$ in $7(x-3)$?
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$7$. The coefficient is the number multiplying the grouped expression.
$7$. The coefficient is the number multiplying the grouped expression.
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What is the factor in $P(1+r)^n$ that does not depend on $P$?
What is the factor in $P(1+r)^n$ that does not depend on $P$?
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$(1+r)^n$. This is the growth/decay factor that remains constant for any value of $P$.
$(1+r)^n$. This is the growth/decay factor that remains constant for any value of $P$.
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What does it mean to treat part of an expression as a single entity in $A(B+C)^2$?
What does it mean to treat part of an expression as a single entity in $A(B+C)^2$?
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View $(B+C)$ as one unit, like a single variable. By grouping terms in parentheses, we can treat complex expressions as simple units.
View $(B+C)$ as one unit, like a single variable. By grouping terms in parentheses, we can treat complex expressions as simple units.
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What is the entity treated as one unit in $ig(1- rac{2}{x}ig)^4$?
What is the entity treated as one unit in $ig(1- rac{2}{x}ig)^4$?
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$ig(1- rac{2}{x}ig)$. This complex fraction expression forms the base of the fourth power.
$ig(1- rac{2}{x}ig)$. This complex fraction expression forms the base of the fourth power.
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What is the factor independent of $x$ in $x(7)^n$?
What is the factor independent of $x$ in $x(7)^n$?
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$(7)^n$. This constant base raised to power $n$ is independent of $x$.
$(7)^n$. This constant base raised to power $n$ is independent of $x$.
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What is the coefficient of the entity $ (x^2-9) $ in $ \frac{7}{4}(x^2-9) $?
What is the coefficient of the entity $ (x^2-9) $ in $ \frac{7}{4}(x^2-9) $?
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$ \frac{7}{4} $. This fraction is the coefficient multiplying the binomial entity.
$ \frac{7}{4} $. This fraction is the coefficient multiplying the binomial entity.
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What is the single entity being multiplied by itself in $(2x-1)(2x-1)$?
What is the single entity being multiplied by itself in $(2x-1)(2x-1)$?
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$(2x-1)$. This binomial multiplies itself to form the square.
$(2x-1)$. This binomial multiplies itself to form the square.
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Identify the entity you can set as $u$ in $ig((x+1)^2+3ig)^2$.
Identify the entity you can set as $u$ in $ig((x+1)^2+3ig)^2$.
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$u=(x+1)^2+3$. The entire squared trinomial can be treated as one variable.
$u=(x+1)^2+3$. The entire squared trinomial can be treated as one variable.
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What is the coefficient of the entity $(x-2)^5$ in $-11(x-2)^5$?
What is the coefficient of the entity $(x-2)^5$ in $-11(x-2)^5$?
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$-11$. This negative coefficient multiplies the entire fifth power expression.
$-11$. This negative coefficient multiplies the entire fifth power expression.
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Identify the entity in $4-3(2x+5)^2$ that is squared.
Identify the entity in $4-3(2x+5)^2$ that is squared.
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$(2x+5)$. The binomial forms the base of the squared term being subtracted.
$(2x+5)$. The binomial forms the base of the squared term being subtracted.
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What is the factor independent of $P$ in $Pig(1+ rac{r}{12}ig)^{12t}$?
What is the factor independent of $P$ in $Pig(1+ rac{r}{12}ig)^{12t}$?
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$ig(1+ rac{r}{12}ig)^{12t}$. This compound interest factor is independent of the principal $P$.
$ig(1+ rac{r}{12}ig)^{12t}$. This compound interest factor is independent of the principal $P$.
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Identify the growth/decay factor entity in $P(0.8)^t$.
Identify the growth/decay factor entity in $P(0.8)^t$.
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$(0.8)^t$. This exponential represents decay since the base is less than 1.
$(0.8)^t$. This exponential represents decay since the base is less than 1.
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What is the entity treated as one unit in $x^2+6x+9=(x+3)^2$?
What is the entity treated as one unit in $x^2+6x+9=(x+3)^2$?
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$(x+3)$. The perfect square trinomial factors as $(x+3)^2$.
$(x+3)$. The perfect square trinomial factors as $(x+3)^2$.
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Identify the entity in $(3z+1)^2-(3z+1)$ that repeats.
Identify the entity in $(3z+1)^2-(3z+1)$ that repeats.
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$(3z+1)$. This trinomial appears in both terms and can be factored out.
$(3z+1)$. This trinomial appears in both terms and can be factored out.
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Identify the common entity in $(p-4)^2+6(p-4)+9$.
Identify the common entity in $(p-4)^2+6(p-4)+9$.
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$(p-4)$. This binomial appears in all three terms of the perfect square trinomial.
$(p-4)$. This binomial appears in all three terms of the perfect square trinomial.
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What is the entity in $5ig(2- rac{x}{3}ig)^2$ that is squared?
What is the entity in $5ig(2- rac{x}{3}ig)^2$ that is squared?
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$ig(2- rac{x}{3}ig)$. This expression with a fraction forms the base being squared.
$ig(2- rac{x}{3}ig)$. This expression with a fraction forms the base being squared.
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Identify the entity substituted by $u$ in $9u^2-4u$ if $u=(x-6)$.
Identify the entity substituted by $u$ in $9u^2-4u$ if $u=(x-6)$.
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$(x-6)$. The substitution $u = (x-6)$ creates the quadratic in $u$.
$(x-6)$. The substitution $u = (x-6)$ creates the quadratic in $u$.
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Which expression treats $(x+1)$ as a single entity multiplied by $3$?
Which expression treats $(x+1)$ as a single entity multiplied by $3$?
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$3(x+1)$. The coefficient 3 multiplies the binomial as a single unit.
$3(x+1)$. The coefficient 3 multiplies the binomial as a single unit.
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What is the factor independent of $x$ in $(x+2)^3(x-5)^2$?
What is the factor independent of $x$ in $(x+2)^3(x-5)^2$?
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None; both factors depend on $x$. Both exponential expressions contain $x$, making them dependent.
None; both factors depend on $x$. Both exponential expressions contain $x$, making them dependent.
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Identify the entity in $2(5y+3)^0$ that is raised to the $0$ power.
Identify the entity in $2(5y+3)^0$ that is raised to the $0$ power.
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$(5y+3)$. Any expression to the zero power equals 1.
$(5y+3)$. Any expression to the zero power equals 1.
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What is the constant factor in $-0.25(4x-1)^6$?
What is the constant factor in $-0.25(4x-1)^6$?
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$-0.25$. This negative decimal multiplies the entire sixth power expression.
$-0.25$. This negative decimal multiplies the entire sixth power expression.
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What is the entity that can be viewed as one unit in $(x^2+1)(x^2+1)-9$?
What is the entity that can be viewed as one unit in $(x^2+1)(x^2+1)-9$?
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$(x^2+1)$. This trinomial appears twice, forming a perfect square difference.
$(x^2+1)$. This trinomial appears twice, forming a perfect square difference.
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Identify the entity in $7-(k+3)^2$ that is being squared.
Identify the entity in $7-(k+3)^2$ that is being squared.
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$(k+3)$. The binomial is squared and then subtracted from 7.
$(k+3)$. The binomial is squared and then subtracted from 7.
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What is the coefficient of the entity $(2a-b)$ in $12(2a-b)$?
What is the coefficient of the entity $(2a-b)$ in $12(2a-b)$?
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$12$. This number is the coefficient multiplying the binomial entity.
$12$. This number is the coefficient multiplying the binomial entity.
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Identify the entity in $ig( rac{x}{2}+5ig)^2$ that is squared.
Identify the entity in $ig( rac{x}{2}+5ig)^2$ that is squared.
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$ig( rac{x}{2}+5ig)$. This fraction expression forms the base of the square.
$ig( rac{x}{2}+5ig)$. This fraction expression forms the base of the square.
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