Proving and Applying Polynomial Identities - Algebra 2
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What is the factored form of $x^2-49$?
What is the factored form of $x^2-49$?
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$(x-7)(x+7)$. Recognize $49=7^2$ and apply difference of squares.
$(x-7)(x+7)$. Recognize $49=7^2$ and apply difference of squares.
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What is the factored form of $x^3+8$?
What is the factored form of $x^3+8$?
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$(x+2)(x^2-2x+4)$. Recognize $8=2^3$ and apply sum of cubes formula.
$(x+2)(x^2-2x+4)$. Recognize $8=2^3$ and apply sum of cubes formula.
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What is the factored form of $x^3-27$?
What is the factored form of $x^3-27$?
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$(x-3)(x^2+3x+9)$. Recognize $27=3^3$ and apply difference of cubes formula.
$(x-3)(x^2+3x+9)$. Recognize $27=3^3$ and apply difference of cubes formula.
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What is the expanded form of $(x+y)^2-(x-y)^2$?
What is the expanded form of $(x+y)^2-(x-y)^2$?
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$4xy$. Difference of squares: $(x+y)^2-(x-y)^2 = 4xy$.
$4xy$. Difference of squares: $(x+y)^2-(x-y)^2 = 4xy$.
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What is the factored form of $x^2-2xy+y^2$?
What is the factored form of $x^2-2xy+y^2$?
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$(x-y)^2$. Perfect square trinomial with negative middle term.
$(x-y)^2$. Perfect square trinomial with negative middle term.
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What is the factored form of $x^2+2xy+y^2$?
What is the factored form of $x^2+2xy+y^2$?
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$(x+y)^2$. Perfect square trinomial with all positive terms.
$(x+y)^2$. Perfect square trinomial with all positive terms.
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What condition on $m$ and $n$ guarantees $a=m^2-n^2$ is positive?
What condition on $m$ and $n$ guarantees $a=m^2-n^2$ is positive?
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$m>n$. Ensures the first leg $a$ is positive in the formula.
$m>n$. Ensures the first leg $a$ is positive in the formula.
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What is $a^2+b^2$ if $a=m^2-n^2$ and $b=2mn$?
What is $a^2+b^2$ if $a=m^2-n^2$ and $b=2mn$?
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$(m^2+n^2)^2$. The hypotenuse squared in Pythagorean triple formula.
$(m^2+n^2)^2$. The hypotenuse squared in Pythagorean triple formula.
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What is $(m^2-n^2)^2+(2mn)^2$ simplified to one expression?
What is $(m^2-n^2)^2+(2mn)^2$ simplified to one expression?
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$(m^2+n^2)^2$. Direct application of the Pythagorean identity $(x^2+y^2)^2$.
$(m^2+n^2)^2$. Direct application of the Pythagorean identity $(x^2+y^2)^2$.
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What identity verifies $(m^2-n^2)^2+(2mn)^2=(m^2+n^2)^2$?
What identity verifies $(m^2-n^2)^2+(2mn)^2=(m^2+n^2)^2$?
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$(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$. The identity that validates all Pythagorean triple formulas.
$(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$. The identity that validates all Pythagorean triple formulas.
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What triple results from $m=4$ and $n=3$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?
What triple results from $m=4$ and $n=3$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?
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$(7,24,25)$. Calculate: $a=16-9=7$, $b=24$, $c=16+9=25$.
$(7,24,25)$. Calculate: $a=16-9=7$, $b=24$, $c=16+9=25$.
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What triple results from $m=4$ and $n=1$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?
What triple results from $m=4$ and $n=1$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?
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$(15,8,17)$. Calculate: $a=16-1=15$, $b=8$, $c=16+1=17$.
$(15,8,17)$. Calculate: $a=16-1=15$, $b=8$, $c=16+1=17$.
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What triple results from $m=3$ and $n=2$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?
What triple results from $m=3$ and $n=2$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?
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$(5,12,13)$. Calculate: $a=9-4=5$, $b=12$, $c=9+4=13$.
$(5,12,13)$. Calculate: $a=9-4=5$, $b=12$, $c=9+4=13$.
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What triple results from $m=2$ and $n=1$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?
What triple results from $m=2$ and $n=1$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?
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$(3,4,5)$. Calculate: $a=4-1=3$, $b=4$, $c=4+1=5$.
$(3,4,5)$. Calculate: $a=4-1=3$, $b=4$, $c=4+1=5$.
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What is the expanded form of $(x^2-3x)^2$?
What is the expanded form of $(x^2-3x)^2$?
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$x^4-6x^3+9x^2$. Square the binomial $x^2-3x$ using $(a-b)^2$ pattern.
$x^4-6x^3+9x^2$. Square the binomial $x^2-3x$ using $(a-b)^2$ pattern.
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What is the expanded form of $(2xy)^2$?
What is the expanded form of $(2xy)^2$?
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$4x^2y^2$. Square each factor: $(2xy)^2 = 2^2 \cdot x^2 \cdot y^2$.
$4x^2y^2$. Square each factor: $(2xy)^2 = 2^2 \cdot x^2 \cdot y^2$.
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What simplified expression equals $(x^2-y^2)^2+(2xy)^2$?
What simplified expression equals $(x^2-y^2)^2+(2xy)^2$?
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$x^4+2x^2y^2+y^4$. Adding the two expressions equals $(x^2+y^2)^2$.
$x^4+2x^2y^2+y^4$. Adding the two expressions equals $(x^2+y^2)^2$.
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What identity results from subtracting $(a-b)^2$ from $(a+b)^2$?
What identity results from subtracting $(a-b)^2$ from $(a+b)^2$?
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$(a+b)^2-(a-b)^2=4ab$. Difference of perfect squares yields four times the product.
$(a+b)^2-(a-b)^2=4ab$. Difference of perfect squares yields four times the product.
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What identity results from adding $(a+b)^2$ and $(a-b)^2$?
What identity results from adding $(a+b)^2$ and $(a-b)^2$?
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$(a+b)^2+(a-b)^2=2(a^2+b^2)$. Sum of perfect squares yields twice the sum of squares.
$(a+b)^2+(a-b)^2=2(a^2+b^2)$. Sum of perfect squares yields twice the sum of squares.
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What is the expanded form of $(x+3)^2$?
What is the expanded form of $(x+3)^2$?
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$x^2+6x+9$. Apply $(a+b)^2$ with $a=x$ and $b=3$.
$x^2+6x+9$. Apply $(a+b)^2$ with $a=x$ and $b=3$.
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What is the expanded form of $(2x-5)^2$?
What is the expanded form of $(2x-5)^2$?
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$4x^2-20x+25$. Apply $(a-b)^2$ with $a=2x$ and $b=5$.
$4x^2-20x+25$. Apply $(a-b)^2$ with $a=2x$ and $b=5$.
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What is the expanded form of $(x-4)(x+4)$?
What is the expanded form of $(x-4)(x+4)$?
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$x^2-16$. Apply difference of squares: $a^2-b^2$ with $a=x$, $b=4$.
$x^2-16$. Apply difference of squares: $a^2-b^2$ with $a=x$, $b=4$.
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What is the expanded form of $(x+2)^3$?
What is the expanded form of $(x+2)^3$?
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$x^3+6x^2+12x+8$. Apply $(a+b)^3$ formula with $a=x$ and $b=2$.
$x^3+6x^2+12x+8$. Apply $(a+b)^3$ formula with $a=x$ and $b=2$.
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What is the identity for $(a-b)(a^2+ab+b^2)$?
What is the identity for $(a-b)(a^2+ab+b^2)$?
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$(a-b)(a^2+ab+b^2)=a^3-b^3$. Difference of cubes factorization produces the cubic expansion.
$(a-b)(a^2+ab+b^2)=a^3-b^3$. Difference of cubes factorization produces the cubic expansion.
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What is the identity for $(a+b)(a^2-ab+b^2)$?
What is the identity for $(a+b)(a^2-ab+b^2)$?
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$(a+b)(a^2-ab+b^2)=a^3+b^3$. Sum of cubes factorization produces the cubic expansion.
$(a+b)(a^2-ab+b^2)=a^3+b^3$. Sum of cubes factorization produces the cubic expansion.
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What is the expanded form of $(x+y)^2+(x-y)^2$?
What is the expanded form of $(x+y)^2+(x-y)^2$?
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$2x^2+2y^2$. Sum of squares: $(x+y)^2+(x-y)^2 = 2(x^2+y^2)$.
$2x^2+2y^2$. Sum of squares: $(x+y)^2+(x-y)^2 = 2(x^2+y^2)$.
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What is the Pythagorean-triple formula from $m$ and $n$: $a,b,c$?
What is the Pythagorean-triple formula from $m$ and $n$: $a,b,c$?
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$a=m^2-n^2,;b=2mn,;c=m^2+n^2$. Standard parametric form generating all primitive Pythagorean triples.
$a=m^2-n^2,;b=2mn,;c=m^2+n^2$. Standard parametric form generating all primitive Pythagorean triples.
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What perfect-square trinomial equals $x^2-12x+36$?
What perfect-square trinomial equals $x^2-12x+36$?
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$(x-6)^2$. Recognize pattern: $2 cdot 6 = 12$ confirms perfect square.
$(x-6)^2$. Recognize pattern: $2 cdot 6 = 12$ confirms perfect square.
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What perfect-square trinomial equals $x^2+10x+25$?
What perfect-square trinomial equals $x^2+10x+25$?
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$(x+5)^2$. Recognize pattern: $2 cdot 5 = 10$ confirms perfect square.
$(x+5)^2$. Recognize pattern: $2 cdot 5 = 10$ confirms perfect square.
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What is the expanded form of $(5x-1)^2$?
What is the expanded form of $(5x-1)^2$?
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$25x^2-10x+1$. Apply formula with $a=5$, $b=-1$.
$25x^2-10x+1$. Apply formula with $a=5$, $b=-1$.
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