Proving and Applying Polynomial Identities - Algebra 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Study these flashcards