Algebra 2 Flashcards: Proving And Applying Polynomial Identities

What this deck covers

This deck focuses on Proving And Applying Polynomial Identities, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the factored form of $x^3-27$?

Answer: $(x-3)(x^2+3x+9)$. Recognize $27=3^3$ and apply difference of cubes formula.

Flashcard 2: What triple results from $m=4$ and $n=3$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?

Answer: $(7,24,25)$. Calculate: $a=16-9=7$, $b=24$, $c=16+9=25$.

Flashcard 3: What identity verifies $(m^2-n^2)^2+(2mn)^2=(m^2+n^2)^2$?

Answer: $(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$. The identity that validates all Pythagorean triple formulas.

Flashcard 4: What identity factors a difference of squares: $a^2-b^2$?

Answer: $a^2-b^2=(a-b)(a+b)$. Fundamental difference of squares factorization pattern.

Flashcard 5: What is the expanded form of $(2x-5)^2$?

Answer: $4x^2-20x+25$. Apply $(a-b)^2$ with $a=2x$ and $b=5$.

Flashcard 6: What is the identity for $(a-b)(a^2+ab+b^2)$?

Answer: $(a-b)(a^2+ab+b^2)=a^3-b^3$. Difference of cubes factorization produces the cubic expansion.

Flashcard 7: What is the Pythagorean-triple formula from $m$ and $n$: $a,b,c$?

Answer: $a=m^2-n^2,\;b=2mn,\;c=m^2+n^2$. Standard parametric form generating all primitive Pythagorean triples.

Flashcard 8: What is the expanded form of $(x+y)^2+(x-y)^2$?

Answer: $2x^2+2y^2$. Sum of squares: $(x+y)^2+(x-y)^2 = 2(x^2+y^2)$.

Flashcard 9: What is the expanded form of $(x^2 + y^2)^2$?

Answer: $x^4+2x^2y^2+y^4$. Square the binomial using $(a+b)^2$ pattern.

Flashcard 10: What is the identity for $(a+b)(a^2-ab+b^2)$?

Answer: $(a+b)(a^2-ab+b^2)=a^3+b^3$. Sum of cubes factorization produces the cubic expansion.

Flashcard 11: What is the factored form of $x^3+8$?

Answer: $(x+2)(x^2-2x+4)$. Recognize $8=2^3$ and apply sum of cubes formula.

Flashcard 12: What is the factored form of $x^4-16$ over the integers?

Answer: $(x-2)(x+2)(x^2+4)$. Apply fourth power difference: $16=2^4$ factorization.

Flashcard 13: What is the identity used to complete the square for $x^2+bx$?

Answer: $x^2+bx=(x+\frac{b}{2})^2-\frac{b^2}{4}$. Complete square by adding and subtracting $(\frac{b}{2})^2$.

Flashcard 14: What is the expanded form of $(2xy)^2$?

Answer: $4x^2y^2$. Square each factor: $(2xy)^2 = 2^2 \cdot x^2 \cdot y^2$.

Flashcard 15: What identity results from adding $(a+b)^2$ and $(a-b)^2$?

Answer: $(a+b)^2+(a-b)^2=2(a^2+b^2)$. Sum of perfect squares yields twice the sum of squares.

Flashcard 16: What is the expanded form of $(x^2-y^2)^2$?

Answer: $x^4-2x^2y^2+y^4$. Square the binomial using $(a-b)^2$ pattern.

Flashcard 17: What perfect-square trinomial equals $x^2-12x+36$?

Answer: $(x-6)^2$. Recognize pattern: $2 cdot 6 = 12$ confirms perfect square.

Flashcard 18: What identity shows $(x^2+y^2)^2$ as a sum of two squares?

Answer: $(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$. Shows how to express fourth power sum as Pythagorean identity.

Flashcard 19: What triple results from $m=4$ and $n=1$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?

Answer: $(15,8,17)$. Calculate: $a=16-1=15$, $b=8$, $c=16+1=17$.

Flashcard 20: What condition on $m$ and $n$ guarantees $a=m^2-n^2$ is positive?

Answer: $m>n$. Ensures the first leg $a$ is positive in the formula.

Flashcard 21: What identity expands a cube of a difference: $(a-b)^3$?

Answer: $(a-b)^3=a^3-3a^2b+3ab^2-b^3$. Binomial cube expansion with alternating negative signs.

Flashcard 22: What is $a^2+b^2$ if $a=m^2-n^2$ and $b=2mn$?

Answer: $(m^2+n^2)^2$. The hypotenuse squared in Pythagorean triple formula.

Flashcard 23: What is the factored form of $x^4-81$ over the integers?

Answer: $(x-3)(x+3)(x^2+9)$. Apply fourth power difference: $81=3^4$ factorization.

Flashcard 24: What identity expands the square of a difference: $(a-b)^2$?

Answer: $(a-b)^2=a^2-2ab+b^2$. Perfect square trinomial with negative middle term $-2ab$.

Flashcard 25: What is the expanded form of $(x+3)^2$?

Answer: $x^2+6x+9$. Apply $(a+b)^2$ with $a=x$ and $b=3$.

Flashcard 26: What identity factors a difference of cubes: $a^3-b^3$?

Answer: $a^3-b^3=(a-b)(a^2+ab+b^2)$. Difference of cubes factors with opposite middle sign.

Flashcard 27: What is the expanded form of $(x^2-3x)^2$?

Answer: $x^4-6x^3+9x^2$. Square the binomial $x^2-3x$ using $(a-b)^2$ pattern.

Flashcard 28: What is $(m^2-n^2)^2+(2mn)^2$ simplified to one expression?

Answer: $(m^2+n^2)^2$. Direct application of the Pythagorean identity $(x^2+y^2)^2$.

Flashcard 29: What is the expanded form of $(x-1)^3$?

Answer: $x^3-3x^2+3x-1$. Apply $(a-b)^3$ formula with $a=x$ and $b=1$.

Flashcard 30: What is the factored form of $x^2+2xy+y^2$?

Answer: $(x+y)^2$. Perfect square trinomial with all positive terms.

Flashcard 31: What identity factors a sum of cubes: $a^3+b^3$?

Answer: $a^3+b^3=(a+b)(a^2-ab+b^2)$. Sum of cubes factors into linear and quadratic terms.

Flashcard 32: What is the factored form of $x^2-2xy+y^2$?

Answer: $(x-y)^2$. Perfect square trinomial with negative middle term.

Flashcard 33: What identity expands a cube of a sum: $(a+b)^3$?

Answer: $(a+b)^3=a^3+3a^2b+3ab^2+b^3$. Binomial cube expansion with alternating signs in coefficients.

Flashcard 34: What is the expanded form of $(3x+2)^2$?

Answer: $9x^2+12x+4$. Apply formula with $a=3$, $b=2$.

Flashcard 35: What is the identity for squaring a binomial with coefficient: $(ax+b)^2$?

Answer: $(ax+b)^2=a^2x^2+2abx+b^2$. General form with coefficient $a$ and constant $b$.

Flashcard 36: What is the expanded form of $(x-4)(x+4)$?

Answer: $x^2-16$. Apply difference of squares: $a^2-b^2$ with $a=x$, $b=4$.

Flashcard 37: What triple results from $m=3$ and $n=2$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?

Answer: $(5,12,13)$. Calculate: $a=9-4=5$, $b=12$, $c=9+4=13$.

Flashcard 38: What is the factored form of $x^2-49$?

Answer: $(x-7)(x+7)$. Recognize $49=7^2$ and apply difference of squares.

Flashcard 39: What simplified expression equals $(x^2-y^2)^2+(2xy)^2$?

Answer: $x^4+2x^2y^2+y^4$. Adding the two expressions equals $(x^2+y^2)^2$.

Flashcard 40: What perfect-square trinomial equals $x^2+10x+25$?

Answer: $(x+5)^2$. Recognize pattern: $2 cdot 5 = 10$ confirms perfect square.

Flashcard 41: What identity expands the square of a sum: $(a+b)^2$?

Answer: $(a+b)^2=a^2+2ab+b^2$. Standard perfect square trinomial form with middle term $2ab$.

Flashcard 42: What is the expanded form of $(x+1)(x^2-x+1)$?

Answer: $x^3+1$. Apply sum of cubes formula: $(a+b)(a^2-ab+b^2)=a^3+b^3$.

Flashcard 43: What is the expanded form of $(x+2)^3$?

Answer: $x^3+6x^2+12x+8$. Apply $(a+b)^3$ formula with $a=x$ and $b=2$.

Flashcard 44: What is the expanded form of $(5x-1)^2$?

Answer: $25x^2-10x+1$. Apply formula with $a=5$, $b=-1$.

Flashcard 45: What is the expanded form of $(x+y)^2-(x-y)^2$?

Answer: $4xy$. Difference of squares: $(x+y)^2-(x-y)^2 = 4xy$.

Flashcard 46: What is the expanded form of $(x-1)(x^2+x+1)$?

Answer: $x^3-1$. Apply difference of cubes formula: $(a-b)(a^2+ab+b^2)=a^3-b^3$.

Flashcard 47: What is the identity for difference of fourth powers: $a^4-b^4$?

Answer: $a^4-b^4=(a-b)(a+b)(a^2+b^2)$. Factor as difference of squares twice: $(a^2)^2-(b^2)^2$.

Flashcard 48: What identity results from subtracting $(a-b)^2$ from $(a+b)^2$?

Answer: $(a+b)^2-(a-b)^2=4ab$. Difference of perfect squares yields four times the product.

Flashcard 49: What is the expanded form of $(x^2+3x)^2$?

Answer: $x^4+6x^3+9x^2$. Square the binomial $x^2+3x$ using $(a+b)^2$ pattern.

Flashcard 50: What is the expanded form of $(x^2+y^2)^2$?

Answer: $x^4+2x^2y^2+y^4$. Square the binomial using $(a+b)^2$ pattern.

Flashcard 51: What triple results from $m=2$ and $n=1$ using $a=m^2-n^2$, $b=2mn$, $c=m^2+n^2$?

Answer: $(3,4,5)$. Calculate: $a=4-1=3$, $b=4$, $c=4+1=5$.