Using Structure to Rewrite Expressions - Algebra 1

1. What is the greatest common factor rewrite for $ax+ay$?

   $ax+ay=a(x+y)$. Factors out the common factor $a$.

2. Factor using sum of cubes: $8y^3+125$.

   $(2y+5)(4y^2-10y+25)$. Recognizes $(2y)^3+5^3$ as a sum of cubes.

3. Factor completely using structure: $x^2+14x+49$.

   $(x+7)^2$. Perfect square trinomial with $a=x$, $b=7$.

4. Rewrite by factoring out the greatest common factor: $15y^2+20y$.

   $5y(3y+4)$. Greatest common factor is $5y$.

5. Factor using sum of cubes: $x^3+27$.

   $(x+3)(x^2-3x+9)$. Recognizes $x^3+3^3$ as a sum of cubes.

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

   $a^3+b^3=(a+b)(a^2-ab+b^2)$. Uses the sum of cubes factoring formula.

7. Factor completely by using structure: $x^4-1$.

   $(x-1)(x+1)(x^2+1)$. Factors $(x^2)^2-1^2$ then continues factoring $x^2-1$.

8. Rewrite by using structure to factor: $x^2-(y+1)^2$.

   $(x-(y+1))(x+(y+1))$. Treats $(y+1)$ as a single term in difference of squares.

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

   $a^2-2ab+b^2=(a-b)^2$. Square of a binomial difference pattern.

10. What is the exponent rule rewrite for $x^m\cdot x^n$?

    $x^m\cdot x^n=x^{m+n}$. Add exponents when multiplying same bases.

11. Rewrite by using exponent structure: $(x^2)^3$.

    $x^6$. Power of a power rule: $(x^2)^3=x^{2\cdot 3}$.

12. Rewrite by using exponent structure: $(2x^3)^2$.

    $4x^6$. Power of a product: $(2x^3)^2=2^2\cdot (x^3)^2$.

13. Factor completely using repeated difference of squares: $x^4-16$.

    $(x-2)(x+2)(x^2+4)$. Factor $(x^2-4)$ further as $(x-2)(x+2)$.

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

    $a^2-b^2=(a-b)(a+b)$. Product of sum and difference of the same terms.

15. Rewrite by factoring using a common binomial: $x(x-3)+5(x-3)$.

    $(x-3)(x+5)$. Factor out the common binomial $(x-3)$.

16. Rewrite by using structure to factor: $9-(x-2)^2$.

    $(5-x)(x+1)$. Recognizes $3^2-(x-2)^2$ as a difference of squares.

17. Rewrite by using exponent structure: $x^2\cdot x^5$.

    $x^7$. Product rule for exponents: $x^2\cdot x^5=x^{2+5}$.

18. What is the exponent rule rewrite for $x^m\div x^n$ (with $x\neq 0$)?

    $x^m\div x^n=x^{m-n}$. Subtract exponents when dividing same bases.

19. Factor completely using structure: $x^6-y^6$.

    $(x^3-y^3)(x^3+y^3)$. Sees $x^6-y^6$ as $(x^3)^2-(y^3)^2$, a difference of squares.

20. Rewrite by factoring using a common binomial: $(x+2)(x-1)-(x+2)3$.

    $(x+2)(x-4)$. Factor out the common binomial $(x+2)$.

21. Factor completely by recognizing a difference of squares: $49y^2-1$.

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

22. Identify the structure of $x^2-10x+25$ as a single squared binomial.

    $(x-5)^2$. Form $a^2-2ab+b^2$ with $a=x$, $b=5$.

23. Rewrite by factoring out the greatest common factor: $8a^3-12a^2$.

    $4a^2(2a-3)$. Greatest common factor is $4a^2$.

24. Factor completely by using structure: $x^4-16$.

    $(x^2-4)(x^2+4)$. Sees $x^4-16$ as $(x^2)^2-4^2$, a difference of squares.

25. Factor by grouping using structure: $ax+ay+bx+by$.

    $(a+b)(x+y)$. Group terms with common factors $(a+b)$ and $(x+y)$.

26. Factor using difference of cubes: $x^3-8$.

    $(x-2)(x^2+2x+4)$. Recognizes $x^3-2^3$ as a difference of cubes.

27. Factor by grouping using structure: $x^2+5x+2x+10$.

    $(x+5)(x+2)$. Groups $x^2+5x$ and $2x+10$ with common factors.

28. Factor by recognizing a difference of squares: $100-t^2$.

    $(10-t)(10+t)$. Recognizes $10^2-t^2$ as a difference of squares.

29. Rewrite by factoring out the greatest common factor: $-10x^3+5x^2$.

    $5x^2(1-2x)$. Factor out $5x^2$ from both terms.

30. Factor by grouping using structure: $3x^2+6x+2x+4$.

    $(3x+2)(x+2)$. Groups $3x^2+6x$ and $2x+4$ with common factors.

31. Factor completely using structure: $x^2-12x+36$.

    $(x-6)^2$. Perfect square trinomial with $a=x$, $b=6$.

32. Factor using difference of cubes: $64a^3-1$.

    $(4a-1)(16a^2+4a+1)$. Recognizes $(4a)^3-1^3$ as a difference of cubes.

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

    $a^3-b^3=(a-b)(a^2+ab+b^2)$. Uses the difference of cubes factoring formula.

34. Rewrite by using structure to factor: $(x+4)^2-9$.

    $(x+1)(x+7)$. Recognizes $(x+4)^2-3^2$ as a difference of squares.

35. What is the distributive property rewrite for $a(b+c)$?

    $a(b+c)=ab+ac$. Distributes $a$ to both terms in parentheses.

36. Factor by recognizing a difference of squares: $x^2-64$.

    $(x-8)(x+8)$. Recognizes $x^2-8^2$ as a difference of squares.

37. Factor completely by recognizing a difference of squares: $9x^2-25$.

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

38. Identify the common binomial factor in $x(x-3)+5(x-3)$.

    $(x-3)$. Both terms contain the binomial factor $(x-3)$.

39. What is the exponent rule rewrite for $(x^m)^n$?

    $(x^m)^n=x^{mn}$. Multiply exponents when raising a power to a power.

40. Factor by grouping using structure: $2x^2-8x+3x-12$.

    $(2x+3)(x-4)$. Groups $2x^2-8x$ and $3x-12$ with common factors.

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

    $a^2+2ab+b^2=(a+b)^2$. Square of a binomial sum pattern.

42. Identify the structure and factor completely: $x^4-y^4$.

    $(x^2-y^2)(x^2+y^2)$. Recognizes $x^4-y^4$ as $(x^2)^2-(y^2)^2$, a difference of squares.

43. Rewrite by factoring out a negative: $-4x^2+12x-9$.

    $-(2x-3)^2$. Factors out $-1$ to make leading coefficient positive.

44. Factor completely by recognizing a perfect square: $25m^2-30m+9$.

    $(5m-3)^2$. Perfect square trinomial with $a=5m$, $b=3$.

45. Factor completely by recognizing a perfect square: $16p^2+24p+9$.

    $(4p+3)^2$. Perfect square trinomial with $a=4p$, $b=3$.

46. Factor completely by recognizing a difference of squares: $4a^2-81$.

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

47. Rewrite by using structure to factor: $(2x-1)^2-16$.

    $(2x-5)(2x+3)$. Recognizes $(2x-1)^2-4^2$ as a difference of squares.

48. Identify the structure of $x^2+10x+25$ as a single squared binomial.

    $(x+5)^2$. Form $a^2+2ab+b^2$ with $a=x$, $b=5$.

49. Factor completely by using structure: $x^4-2x^2+1$.

    $(x^2-1)^2$. Sees structure as $(x^2-1)^2$ where $u=x^2-1$.

50. Rewrite by using exponent structure: $\frac{x^9}{x^4}$ (with $x\neq 0$).

    $x^5$. Quotient rule for exponents: $x^9\div x^4=x^{9-4}$.

51. Rewrite by factoring out the greatest common factor: $6x^2+9x$.

    $3x(2x+3)$. Greatest common factor is $3x$.

52. Rewrite by factoring out a negative to make the leading term positive: $-x^2+6x-9$.

    $-(x-3)^2$. Factors out $-1$ to make leading coefficient positive.

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