Using Structure to Rewrite Expressions - Algebra
Card 1 of 30
Identify the structure of $x^2-10x+25$ as a single squared binomial.
Identify the structure of $x^2-10x+25$ as a single squared binomial.
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$(x-5)^2$. Form $a^2-2ab+b^2$ with $a=x$, $b=5$.
$(x-5)^2$. Form $a^2-2ab+b^2$ with $a=x$, $b=5$.
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Factor completely using structure: $x^2+14x+49$.
Factor completely using structure: $x^2+14x+49$.
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$(x+7)^2$. Perfect square trinomial with $a=x$, $b=7$.
$(x+7)^2$. Perfect square trinomial with $a=x$, $b=7$.
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What is the exponent rule rewrite for $x^m\div x^n$ (with $x\neq 0$)?
What is the exponent rule rewrite for $x^m\div x^n$ (with $x\neq 0$)?
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$x^m\div x^n=x^{m-n}$. Subtract exponents when dividing same bases.
$x^m\div x^n=x^{m-n}$. Subtract exponents when dividing same bases.
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Rewrite by factoring out the greatest common factor: $6x^2+9x$.
Rewrite by factoring out the greatest common factor: $6x^2+9x$.
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$3x(2x+3)$. Greatest common factor is $3x$.
$3x(2x+3)$. Greatest common factor is $3x$.
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Rewrite by factoring out the greatest common factor: $8a^3-12a^2$.
Rewrite by factoring out the greatest common factor: $8a^3-12a^2$.
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$4a^2(2a-3)$. Greatest common factor is $4a^2$.
$4a^2(2a-3)$. Greatest common factor is $4a^2$.
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Rewrite by factoring out the greatest common factor: $15y^2+20y$.
Rewrite by factoring out the greatest common factor: $15y^2+20y$.
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$5y(3y+4)$. Greatest common factor is $5y$.
$5y(3y+4)$. Greatest common factor is $5y$.
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What is the factoring pattern for a difference of squares $a^2-b^2$?
What is the factoring pattern for a difference of squares $a^2-b^2$?
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$a^2-b^2=(a-b)(a+b)$. Product of sum and difference of the same terms.
$a^2-b^2=(a-b)(a+b)$. Product of sum and difference of the same terms.
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What is the factoring pattern for a perfect square trinomial $a^2+2ab+b^2$?
What is the factoring pattern for a perfect square trinomial $a^2+2ab+b^2$?
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$a^2+2ab+b^2=(a+b)^2$. Square of a binomial sum pattern.
$a^2+2ab+b^2=(a+b)^2$. Square of a binomial sum pattern.
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What is the factoring pattern for a sum of cubes $a^3+b^3$?
What is the factoring pattern for a sum of cubes $a^3+b^3$?
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$a^3+b^3=(a+b)(a^2-ab+b^2)$. Uses the sum of cubes factoring formula.
$a^3+b^3=(a+b)(a^2-ab+b^2)$. Uses the sum of cubes factoring formula.
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Factor completely by using structure: $x^4-16$.
Factor completely by using structure: $x^4-16$.
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$(x^2-4)(x^2+4)$. Sees $x^4-16$ as $(x^2)^2-4^2$, a difference of squares.
$(x^2-4)(x^2+4)$. Sees $x^4-16$ as $(x^2)^2-4^2$, a difference of squares.
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Identify the structure and factor completely: $x^4-y^4$.
Identify the structure and factor completely: $x^4-y^4$.
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$(x^2-y^2)(x^2+y^2)$. Recognizes $x^4-y^4$ as $(x^2)^2-(y^2)^2$, a difference of squares.
$(x^2-y^2)(x^2+y^2)$. Recognizes $x^4-y^4$ as $(x^2)^2-(y^2)^2$, a difference of squares.
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What is the factoring pattern for a perfect square trinomial $a^2-2ab+b^2$?
What is the factoring pattern for a perfect square trinomial $a^2-2ab+b^2$?
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$a^2-2ab+b^2=(a-b)^2$. Square of a binomial difference pattern.
$a^2-2ab+b^2=(a-b)^2$. Square of a binomial difference pattern.
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Factor completely by recognizing a difference of squares: $9x^2-25$.
Factor completely by recognizing a difference of squares: $9x^2-25$.
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$(3x-5)(3x+5)$. Recognizes $(3x)^2-5^2$ as a difference of squares.
$(3x-5)(3x+5)$. Recognizes $(3x)^2-5^2$ as a difference of squares.
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Factor completely by recognizing a difference of squares: $49y^2-1$.
Factor completely by recognizing a difference of squares: $49y^2-1$.
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$(7y-1)(7y+1)$. Recognizes $(7y)^2-1^2$ as a difference of squares.
$(7y-1)(7y+1)$. Recognizes $(7y)^2-1^2$ as a difference of squares.
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Factor completely by recognizing a difference of squares: $4a^2-81$.
Factor completely by recognizing a difference of squares: $4a^2-81$.
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$(2a-9)(2a+9)$. Recognizes $(2a)^2-9^2$ as a difference of squares.
$(2a-9)(2a+9)$. Recognizes $(2a)^2-9^2$ as a difference of squares.
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Factor completely using structure: $x^2-12x+36$.
Factor completely using structure: $x^2-12x+36$.
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$(x-6)^2$. Perfect square trinomial with $a=x$, $b=6$.
$(x-6)^2$. Perfect square trinomial with $a=x$, $b=6$.
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Factor completely using structure: $x^2+14x+49$.
Factor completely using structure: $x^2+14x+49$.
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$(x+7)^2$. Perfect square trinomial with $a=x$, $b=7$.
$(x+7)^2$. Perfect square trinomial with $a=x$, $b=7$.
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Factor completely by recognizing a perfect square: $25m^2-30m+9$.
Factor completely by recognizing a perfect square: $25m^2-30m+9$.
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$(5m-3)^2$. Perfect square trinomial with $a=5m$, $b=3$.
$(5m-3)^2$. Perfect square trinomial with $a=5m$, $b=3$.
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What is the exponent rule rewrite for $x^m\cdot x^n$?
What is the exponent rule rewrite for $x^m\cdot x^n$?
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$x^m\cdot x^n=x^{m+n}$. Add exponents when multiplying same bases.
$x^m\cdot x^n=x^{m+n}$. Add exponents when multiplying same bases.
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What is the distributive property rewrite for $a(b+c)$?
What is the distributive property rewrite for $a(b+c)$?
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$a(b+c)=ab+ac$. Distributes $a$ to both terms in parentheses.
$a(b+c)=ab+ac$. Distributes $a$ to both terms in parentheses.
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What is the greatest common factor rewrite for $ax+ay$?
What is the greatest common factor rewrite for $ax+ay$?
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$ax+ay=a(x+y)$. Factors out the common factor $a$.
$ax+ay=a(x+y)$. Factors out the common factor $a$.
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What is the factoring pattern for a difference of cubes $a^3-b^3$?
What is the factoring pattern for a difference of cubes $a^3-b^3$?
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$a^3-b^3=(a-b)(a^2+ab+b^2)$. Uses the difference of cubes factoring formula.
$a^3-b^3=(a-b)(a^2+ab+b^2)$. Uses the difference of cubes factoring formula.
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Factor completely using repeated difference of squares: $x^4-16$.
Factor completely using repeated difference of squares: $x^4-16$.
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$(x-2)(x+2)(x^2+4)$. Factor $(x^2-4)$ further as $(x-2)(x+2)$.
$(x-2)(x+2)(x^2+4)$. Factor $(x^2-4)$ further as $(x-2)(x+2)$.
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What is the exponent rule rewrite for $(x^m)^n$?
What is the exponent rule rewrite for $(x^m)^n$?
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$(x^m)^n=x^{mn}$. Multiply exponents when raising a power to a power.
$(x^m)^n=x^{mn}$. Multiply exponents when raising a power to a power.
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Identify the common binomial factor in $x(x-3)+5(x-3)$.
Identify the common binomial factor in $x(x-3)+5(x-3)$.
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$(x-3)$. Both terms contain the binomial factor $(x-3)$.
$(x-3)$. Both terms contain the binomial factor $(x-3)$.
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Rewrite by using structure to factor: $(2x-1)^2-16$.
Rewrite by using structure to factor: $(2x-1)^2-16$.
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$(2x-5)(2x+3)$. Recognizes $(2x-1)^2-4^2$ as a difference of squares.
$(2x-5)(2x+3)$. Recognizes $(2x-1)^2-4^2$ as a difference of squares.
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Rewrite by using structure to factor: $(x+4)^2-9$.
Rewrite by using structure to factor: $(x+4)^2-9$.
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$(x+1)(x+7)$. Recognizes $(x+4)^2-3^2$ as a difference of squares.
$(x+1)(x+7)$. Recognizes $(x+4)^2-3^2$ as a difference of squares.
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Rewrite by using structure to factor: $x^2-(y+1)^2$.
Rewrite by using structure to factor: $x^2-(y+1)^2$.
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$(x-(y+1))(x+(y+1))$. Treats $(y+1)$ as a single term in difference of squares.
$(x-(y+1))(x+(y+1))$. Treats $(y+1)$ as a single term in difference of squares.
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Rewrite by using exponent structure: $(2x^3)^2$.
Rewrite by using exponent structure: $(2x^3)^2$.
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$4x^6$. Power of a product: $(2x^3)^2=2^2\cdot (x^3)^2$.
$4x^6$. Power of a product: $(2x^3)^2=2^2\cdot (x^3)^2$.
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Factor by recognizing a difference of squares: $100-t^2$.
Factor by recognizing a difference of squares: $100-t^2$.
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$(10-t)(10+t)$. Recognizes $10^2-t^2$ as a difference of squares.
$(10-t)(10+t)$. Recognizes $10^2-t^2$ as a difference of squares.
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