Proving and Applying Polynomial Identities - Geometry
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What identity shows $x^4-y^4$ factors as a product of two quadratics?
What identity shows $x^4-y^4$ factors as a product of two quadratics?
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$x^4-y^4=(x^2-y^2)(x^2+y^2)$. Difference of fourth powers factorization.
$x^4-y^4=(x^2-y^2)(x^2+y^2)$. Difference of fourth powers factorization.
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What is the fully factored form of $x^4-y^4$ over the reals?
What is the fully factored form of $x^4-y^4$ over the reals?
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$x^4-y^4=(x-y)(x+y)(x^2+y^2)$. Complete factorization over real numbers.
$x^4-y^4=(x-y)(x+y)(x^2+y^2)$. Complete factorization over real numbers.
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What is the identity that factors $x^2+2xy+y^2$?
What is the identity that factors $x^2+2xy+y^2$?
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$x^2+2xy+y^2=(x+y)^2$. Perfect square trinomial pattern.
$x^2+2xy+y^2=(x+y)^2$. Perfect square trinomial pattern.
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What is the identity that factors $x^2-2xy+y^2$?
What is the identity that factors $x^2-2xy+y^2$?
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$x^2-2xy+y^2=(x-y)^2$. Perfect square trinomial pattern.
$x^2-2xy+y^2=(x-y)^2$. Perfect square trinomial pattern.
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What is the expansion of $(x^2+y^2)^2$?
What is the expansion of $(x^2+y^2)^2$?
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$x^4+2x^2y^2+y^4$. Perfect square expansion formula.
$x^4+2x^2y^2+y^4$. Perfect square expansion formula.
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What is the expansion of $(2xy)^2$?
What is the expansion of $(2xy)^2$?
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$4x^2y^2$. Square the coefficient: $(2)^2=4$.
$4x^2y^2$. Square the coefficient: $(2)^2=4$.
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What is the simplified result of $(m^2+n^2)^2-(2mn)^2$?
What is the simplified result of $(m^2+n^2)^2-(2mn)^2$?
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$(m^2-n^2)^2$. Subtract $(2mn)^2$ from expansion.
$(m^2-n^2)^2$. Subtract $(2mn)^2$ from expansion.
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Identify the missing expression: $(m^2-n^2)^2+(2mn)^2=\square$.
Identify the missing expression: $(m^2-n^2)^2+(2mn)^2=\square$.
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$(m^2+n^2)^2$. Euclid's formula verification identity.
$(m^2+n^2)^2$. Euclid's formula verification identity.
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Find and correct the error: $(x^2+y^2)^2=x^4+2xy+y^4$.
Find and correct the error: $(x^2+y^2)^2=x^4+2xy+y^4$.
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Correct: $(x^2+y^2)^2=x^4+2x^2y^2+y^4$. Missing $x^2y^2$ terms in middle.
Correct: $(x^2+y^2)^2=x^4+2x^2y^2+y^4$. Missing $x^2y^2$ terms in middle.
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What is the simplified result of $(m^2+n^2)+(2mn)$?
What is the simplified result of $(m^2+n^2)+(2mn)$?
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$(m+n)^2$. Add to form perfect square.
$(m+n)^2$. Add to form perfect square.
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What is the value of $(m^2-n^2)^2+(2mn)^2$ simplified?
What is the value of $(m^2-n^2)^2+(2mn)^2$ simplified?
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$(m^2+n^2)^2$. Identity confirms Pythagorean relationship.
$(m^2+n^2)^2$. Identity confirms Pythagorean relationship.
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What is the identity that factors $x^2+2xy+y^2$?
What is the identity that factors $x^2+2xy+y^2$?
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$x^2+2xy+y^2=(x+y)^2$. Perfect square trinomial pattern.
$x^2+2xy+y^2=(x+y)^2$. Perfect square trinomial pattern.
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Find and correct the error: $(x^2-y^2)^2=x^4-2xy+y^4$.
Find and correct the error: $(x^2-y^2)^2=x^4-2xy+y^4$.
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Correct: $(x^2-y^2)^2=x^4-2x^2y^2+y^4$. Missing $x^2y^2$ terms in middle.
Correct: $(x^2-y^2)^2=x^4-2x^2y^2+y^4$. Missing $x^2y^2$ terms in middle.
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What is the simplified result of $(m^2+n^2)^2-(m^2-n^2)^2$?
What is the simplified result of $(m^2+n^2)^2-(m^2-n^2)^2$?
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$4m^2n^2$. Difference of squares gives $4m^2n^2$.
$4m^2n^2$. Difference of squares gives $4m^2n^2$.
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What is the simplified result of $(x+y)^2+(x-y)^2$?
What is the simplified result of $(x+y)^2+(x-y)^2$?
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$2x^2+2y^2$. Cross terms cancel, leaving doubled squares.
$2x^2+2y^2$. Cross terms cancel, leaving doubled squares.
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What is the simplified result of $(m^2-n^2)(m^2+n^2)$?
What is the simplified result of $(m^2-n^2)(m^2+n^2)$?
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$m^4-n^4$. Difference of squares formula applied.
$m^4-n^4$. Difference of squares formula applied.
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What is the difference of squares identity for $ (a+b)(a-b) $?
What is the difference of squares identity for $ (a+b)(a-b) $?
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$ (a+b)(a-b) = a^2 - b^2 $. Middle terms cancel: $ ab - ab = 0 $.
$ (a+b)(a-b) = a^2 - b^2 $. Middle terms cancel: $ ab - ab = 0 $.
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What is the simplified result of $ (m^2 + n^2)^2 - (2mn)^2 $?
What is the simplified result of $ (m^2 + n^2)^2 - (2mn)^2 $?
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$ (m^2 - n^2)^2 $. Subtract $ (2mn)^2 $ from expansion.
$ (m^2 - n^2)^2 $. Subtract $ (2mn)^2 $ from expansion.
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What is the identity for $a^2-b^2$ factored completely over the reals?
What is the identity for $a^2-b^2$ factored completely over the reals?
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$a^2-b^2=(a-b)(a+b)$. Factors as conjugate pair multiplication.
$a^2-b^2=(a-b)(a+b)$. Factors as conjugate pair multiplication.
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What is the identity for $a^3-b^3$ factored?
What is the identity for $a^3-b^3$ factored?
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$a^3-b^3=(a-b)(a^2+ab+b^2)$. Difference of cubes factorization formula.
$a^3-b^3=(a-b)(a^2+ab+b^2)$. Difference of cubes factorization formula.
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What identity rewrites $(x^2+y^2)^2$ in terms of $(x^2-y^2)$ and $2xy$?
What identity rewrites $(x^2+y^2)^2$ in terms of $(x^2-y^2)$ and $2xy$?
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$(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$. Generates Pythagorean triples when expanded.
$(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$. Generates Pythagorean triples when expanded.
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In $(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$, what is the common hypotenuse expression?
In $(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$, what is the common hypotenuse expression?
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$x^2+y^2$. Represents the hypotenuse in Pythagorean context.
$x^2+y^2$. Represents the hypotenuse in Pythagorean context.
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In Euclid’s formula, what are the legs in terms of integers $m$ and $n$ with $m>n$?
In Euclid’s formula, what are the legs in terms of integers $m$ and $n$ with $m>n$?
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$a=m^2-n^2$ and $b=2mn$. Generates the two legs of the triangle.
$a=m^2-n^2$ and $b=2mn$. Generates the two legs of the triangle.
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In Euclid’s formula, what is the hypotenuse in terms of integers $m$ and $n$?
In Euclid’s formula, what is the hypotenuse in terms of integers $m$ and $n$?
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$c=m^2+n^2$. Formula for the hypotenuse length.
$c=m^2+n^2$. Formula for the hypotenuse length.
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Which condition on $m$ and $n$ guarantees a primitive Pythagorean triple from Euclid’s formula?
Which condition on $m$ and $n$ guarantees a primitive Pythagorean triple from Euclid’s formula?
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$ ext{gcd}(m,n)=1$ and $m,n$ have opposite parity. Ensures the triple has no common factors.
$ ext{gcd}(m,n)=1$ and $m,n$ have opposite parity. Ensures the triple has no common factors.
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What is the parity fact for Euclid’s formula when $m$ and $n$ have opposite parity?
What is the parity fact for Euclid’s formula when $m$ and $n$ have opposite parity?
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One leg is even and the other leg is odd. Opposite parity ensures one even, one odd leg.
One leg is even and the other leg is odd. Opposite parity ensures one even, one odd leg.
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What is the key numerical relationship described by a Pythagorean triple $(a,b,c)$?
What is the key numerical relationship described by a Pythagorean triple $(a,b,c)$?
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$a^2+b^2=c^2$. Pythagorean theorem relationship.
$a^2+b^2=c^2$. Pythagorean theorem relationship.
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Identify the identity: $(x+y)^2-(x-y)^2$ equals what simplified expression?
Identify the identity: $(x+y)^2-(x-y)^2$ equals what simplified expression?
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$4xy$. Expansion gives $4xy$ after simplification.
$4xy$. Expansion gives $4xy$ after simplification.
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What is the expansion of $(x^2-y^2)^2$?
What is the expansion of $(x^2-y^2)^2$?
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$x^4-2x^2y^2+y^4$. Perfect square with negative middle term.
$x^4-2x^2y^2+y^4$. Perfect square with negative middle term.
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What is the expansion of $(2xy)^2$?
What is the expansion of $(2xy)^2$?
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$4x^2y^2$. Square the coefficient: $(2)^2=4$.
$4x^2y^2$. Square the coefficient: $(2)^2=4$.
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