Using Structure to Rewrite Expressions - Algebra 2

1. What is $6a^2b+9ab^2$ rewritten by factoring out the GCF?

   $3ab(2a+3b)$. Factor out common term $3ab$.

2. What is $x^2-6x+9$ rewritten as a square?

   $(x-3)^2$. Recognizes $x^2-2(x)(3)+3^2$ as perfect square trinomial.

3. What is the factoring pattern for a perfect square trinomial $a^2-2ab+b^2$?

   $a^2-2ab+b^2=(a-b)^2$. First term squared minus twice the product plus second term squared.

4. What is $2x^2+8x$ rewritten by factoring out the GCF?

   $2x(x+4)$. Factor out common term $2x$.

5. What is $x^2-9y^2$ factored using structure?

   $(x-3y)(x+3y)$. Recognizes $x^2-(3y)^2$ as difference of squares.

6. What is the factoring pattern for a difference of squares $a^2-b^2$?

   $a^2-b^2=(a-b)(a+b)$. Standard pattern for difference of two perfect squares.

7. What is $x^3+y^3$ factored using structure?

   $(x+y)(x^2-xy+y^2)$. Standard sum of cubes formula.

8. What is $9x^2-25$ factored using structure?

   $(3x-5)(3x+5)$. Recognizes $(3x)^2-5^2$ as difference of squares.

9. What is $8x^3-1$ factored using structure?

   $(2x-1)(4x^2+2x+1)$. Recognizes $(2x)^3-1^3$ and applies difference of cubes formula.

10. What is $16a^2b^2-81$ factored using structure?

    $(4ab-9)(4ab+9)$. Recognizes $(4ab)^2-9^2$ as difference of squares.

11. What is $x^4-2x^2+1$ rewritten using structure?

    $(x^2-1)^2$. Recognizes $(x^2)^2-2(x^2)(1)+1^2$ as perfect square trinomial.

12. What is $9w^2+24w+16$ rewritten as a square?

    $(3w+4)^2$. Recognizes $(3w)^2+2(3w)(4)+4^2$ as perfect square trinomial.

13. What is $x^2+2xy+y^2$ rewritten using structure?

    $(x+y)^2$. Standard perfect square trinomial pattern.

14. What is $x^2+7x+12$ factored using structure?

    $(x+3)(x+4)$. Find two numbers that multiply to $12$ and add to $7$: $3$ and $4$.

15. What is $x^3+8$ factored using structure?

    $(x+2)(x^2-2x+4)$. Recognizes $x^3+2^3$ and applies sum of cubes formula.

16. What is $x^2-5x+6$ factored using structure?

    $(x-2)(x-3)$. Find two numbers that multiply to $6$ and add to $-5$: $-2$ and $-3$.

17. What is $x^4-6x^2+9$ rewritten using structure?

    $(x^2-3)^2$. Recognizes $(x^2)^2-2(x^2)(3)+3^2$ as perfect square trinomial.

18. What is $x^4-9$ factored completely over the integers?

    $(x^2-3)(x^2+3)$. Recognizes $(x^2)^2-3^2$ as difference of squares.

19. What is $x^3-y^3$ factored using structure?

    $(x-y)(x^2+xy+y^2)$. Standard difference of cubes formula.

20. What is $x^2+10x+25$ rewritten as a perfect square?

    $(x+5)^2$. Recognizes $x^2+2(x)(5)+5^2$ as perfect square trinomial.

21. What is $x^2-2xy+y^2$ rewritten using structure?

    $(x-y)^2$. Standard perfect square trinomial pattern.

22. What structure suggests factoring by grouping in a $4$-term polynomial?

    Two pairs of terms with a common binomial factor. Group terms in pairs to reveal a common binomial factor.

23. What is $4x^2-12x+9$ rewritten as a perfect square?

    $(2x-3)^2$. Recognizes $(2x)^2-2(2x)(3)+3^2$ as perfect square trinomial.

24. What is $18y^2-8$ factored completely using GCF and structure?

    $2(3y-2)(3y+2)$. Factor out GCF of $2$, then recognize difference of squares.

25. What is $4x^2-y^2$ factored using structure?

    $(2x-y)(2x+y)$. Recognizes $(2x)^2-y^2$ as difference of squares.

26. What is $25p^2-30p+9$ rewritten as a perfect square?

    $(5p-3)^2$. Recognizes $(5p)^2-2(5p)(3)+3^2$ as perfect square trinomial.

27. What is the factoring pattern for a perfect square trinomial $a^2+2ab+b^2$?

    $a^2+2ab+b^2=(a+b)^2$. First term squared plus twice the product plus second term squared.

28. What is $4x^2+12x+9$ rewritten as a perfect square?

    $(2x+3)^2$. Recognizes $(2x)^2+2(2x)(3)+3^2$ as perfect square trinomial.

29. What is $x^4+2x^2+1$ rewritten using structure?

    $(x^2+1)^2$. Recognizes $(x^2)^2+2(x^2)(1)+1^2$ as perfect square trinomial.

30. Identify the rewrite that shows $x^4-y^4$ as a difference of squares.

    $x^4-y^4=(x^2)^2-(y^2)^2$. Recognizes $x^4$ as $(x^2)^2$ and $y^4$ as $(y^2)^2$.

31. What is $2x^2+7x+3$ factored using structure?

    $(2x+1)(x+3)$. Use factoring by grouping or trial method for quadratic.

32. What is $15x^3y-10x^2y^2$ rewritten by factoring out the GCF?

    $5x^2y(3x-2y)$. Factor out common term $5x^2y$.

33. What is $36m^2-49n^2$ factored using structure?

    $(6m-7n)(6m+7n)$. Recognizes $(6m)^2-(7n)^2$ as difference of squares.

34. What is the factoring pattern for a difference of cubes $a^3-b^3$?

    $a^3-b^3=(a-b)(a^2+ab+b^2)$. Standard formula for difference of two perfect cubes.

35. What is $a^4-16$ factored completely over the integers?

    $(a-2)(a+2)(a^2+4)$. Factor as $(a^2)^2-4^2$, then apply difference of squares twice.

36. What is $x^4-1$ factored completely over the integers?

    $(x-1)(x+1)(x^2+1)$. Factor as $(x^2)^2-1^2$, then apply difference of squares twice.

37. What is $49m^2n^2-1$ factored using structure?

    $(7mn-1)(7mn+1)$. Recognizes $(7mn)^2-1^2$ as difference of squares.

38. What is $81x^4-1$ factored completely over the integers?

    $(3x-1)(3x+1)(9x^2+1)$. Factor as $(9x^2)^2-1^2$, then apply difference of squares twice.

39. What common factor should you look for first when rewriting a polynomial expression?

    Greatest common factor (GCF). Always factor out common terms before applying other patterns.

40. What is $y^2-12y+36$ rewritten as a perfect square?

    $(y-6)^2$. Recognizes $y^2-2(y)(6)+6^2$ as perfect square trinomial.

41. What is $x^4-y^4$ factored completely over the integers?

    $(x-y)(x+y)(x^2+y^2)$. Factor difference of squares twice: $(x^2-y^2)(x^2+y^2) = (x-y)(x+y)(x^2+y^2)$.

42. What is $x^2-y^2$ factored using structure?

    $(x-y)(x+y)$. Standard difference of squares pattern.

43. What is $100-4z^2$ factored completely?

    $4(5-z)(5+z)$. Factor out GCF of $4$, then recognize difference of squares.

44. What is $12x^2-27$ factored completely using GCF and structure?

    $3(2x-3)(2x+3)$. Factor out GCF of $3$, then recognize difference of squares.

45. What is the factoring pattern for a sum of cubes $a^3+b^3$?

    $a^3+b^3=(a+b)(a^2-ab+b^2)$. Standard formula for sum of two perfect cubes.

46. What is $64t^3+125$ factored using structure?

    $(4t+5)(16t^2-20t+25)$. Recognizes $(4t)^3+5^3$ and applies sum of cubes formula.

47. What is $27a^3-b^3$ factored using structure?

    $(3a-b)(9a^2+3ab+b^2)$. Recognizes $(3a)^3-b^3$ and applies difference of cubes formula.

48. What is $x^2-81$ factored using structure?

    $(x-9)(x+9)$. Recognizes $x^2-9^2$ as difference of squares.

49. What is $x^4+6x^2+9$ rewritten using structure?

    $(x^2+3)^2$. Recognizes $(x^2)^2+2(x^2)(3)+3^2$ as perfect square trinomial.

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