Extending Polynomial Identities to Complex Numbers - Algebra 2

1. Rewrite $x^2+36$ as a product of two complex conjugate binomials.

   $(x+6i)(x-6i)$. $x^2 + 36 = x^2 + (6i)^2$ factors as conjugate pair.

2. Rewrite $x^2+5$ as a product of two complex conjugate binomials.

   $(x+i\sqrt{5})(x-i\sqrt{5})$. $x^2 + 5 = x^2 + (i\sqrt{5})^2$ factors as conjugate pair.

3. What is $i^2$?

   $i^2=-1$. Definition of the imaginary unit.

4. What is $i^{27}$?

   $i^{27}=-i$. $i^{27} = i^{24} \cdot i^3 = 1 \cdot (-i) = -i$

5. Rewrite $x^2+50$ as a product of two complex conjugate binomials.

   $(x+5i\sqrt{2})(x-5i\sqrt{2})$. $50 = 25 \cdot 2$, so $\sqrt{50} = 5\sqrt{2}$.

6. What is $i^4$?

   $i^4=1$. $i^4 = (i^2)^2 = (-1)^2 = 1$

7. What are the complex roots of $x^2-2x+2=0$?

   $x=1+i,1-i$. Use quadratic formula with discriminant $4 - 8 = -4$.

8. Factor $x^2+2x+5$ over the complex numbers.

   $(x+1+2i)(x+1-2i)$. Complete the square: $(x+1)^2 - 1 + 5 = (x+1)^2 + 4$.

9. What are the complex roots of $x^2+9=0$?

   $x=3i,-3i$. Solve $x^2 = -9$ to get $x = \pm 3i$.

10. Rewrite $x^2+2$ as a product of two complex conjugate binomials.

    $(x+i\sqrt{2})(x-i\sqrt{2})$. $x^2 + 2 = x^2 + (i\sqrt{2})^2$ factors as conjugate pair.

11. What is $i^{100}$?

    $i^{100}=1$. $i^{100} = (i^4)^{25} = 1^{25} = 1$

12. Factor $x^2+4x+8$ over the complex numbers.

    $(x+2+2i)(x+2-2i)$. Complete the square: $(x+2)^2 - 4 + 8 = (x+2)^2 + 4$.

13. Factor $x^2+8x+20$ over the complex numbers.

    $(x+4+2i)(x+4-2i)$. Complete the square: $(x+4)^2 - 16 + 20 = (x+4)^2 + 4$.

14. Identify the factorization of $x^2-2ax+(a^2+b^2)$ over complex numbers.

    $(x-(a+bi))(x-(a-bi))$. Standard form with roots $a \pm bi$.

15. What is $i^3$?

    $i^3=-i$. $i^3 = i^2 \cdot i = -1 \cdot i = -i$

16. Rewrite $x^2+27$ as a product of two complex conjugate binomials.

    $(x+3i\sqrt{3})(x-3i\sqrt{3})$. $27 = 9 \cdot 3$, so $\sqrt{27} = 3\sqrt{3}$.

17. Rewrite $x^2+7$ as a product of two complex conjugate binomials.

    $(x+i\sqrt{7})(x-i\sqrt{7})$. $x^2 + 7 = x^2 + (i\sqrt{7})^2$ factors as conjugate pair.

18. State the identity for factoring a difference of squares over complex numbers.

    $a^2-b^2=(a-b)(a+b)$. Standard difference of squares identity, valid for complex numbers.

19. What are the complex roots of $x^2+1=0$?

    $x=i,-i$. Solve $x^2 = -1$ to get $x = \pm i$.

20. Factor $x^2-6x+13$ over the complex numbers.

    $(x-3+2i)(x-3-2i)$. Complete the square: $(x-3)^2 - 9 + 13 = (x-3)^2 + 4$.

21. What are the complex roots of $x^2-6x+13=0$?

    $x=3+2i,3-2i$. Use quadratic formula with discriminant $36 - 52 = -16$.

22. Rewrite $x^2+64$ as a product of two complex conjugate binomials.

    $(x+8i)(x-8i)$. $x^2 + 64 = x^2 + (8i)^2$ factors as conjugate pair.

23. Rewrite $x^2+12$ as a product of two complex conjugate binomials.

    $(x+2i\sqrt{3})(x-2i\sqrt{3})$. $12 = 4 \cdot 3$, so $\sqrt{12} = 2\sqrt{3}$.

24. Rewrite $x^2+25$ as a product of two complex conjugate binomials.

    $(x+5i)(x-5i)$. $x^2 + 25 = x^2 + (5i)^2$ factors as conjugate pair.

25. What are the complex roots of $x^2+4=0$?

    $x=2i,-2i$. Solve $x^2 = -4$ to get $x = \pm 2i$.

26. Factor $x^2+10x+29$ over the complex numbers.

    $(x+5+2i)(x+5-2i)$. Complete the square: $(x+5)^2 - 25 + 29 = (x+5)^2 + 4$.

27. Factor $x^2-2x+2$ over the complex numbers.

    $(x-1+i)(x-1-i)$. Complete the square: $(x-1)^2 - 1 + 2 = (x-1)^2 + 1$.

28. Factor $x^2-10x+29$ over the complex numbers.

    $(x-5+2i)(x-5-2i)$. Complete the square: $(x-5)^2 - 25 + 29 = (x-5)^2 + 4$.

29. What are the complex roots of $x^2+2x+5=0$?

    $x=-1+2i,-1-2i$. Use quadratic formula with discriminant $4 - 20 = -16$.

30. Rewrite $x^2+1$ as a product of two complex conjugate binomials.

    $(x+i)(x-i)$. $x^2 + 1 = x^2 + i^2$ factors as conjugate pair.

31. What is the factored form of $x^2+2ax+(a^2+b^2)$ over complex numbers?

    $(x-(-a+bi))(x-(-a-bi))$. Standard form with roots $-a \pm bi$.

32. Factor $x^2+6x+13$ over the complex numbers.

    $(x+3+2i)(x+3-2i)$. Complete the square: $(x+3)^2 - 9 + 13 = (x+3)^2 + 4$.

33. Rewrite $x^2+49$ as a product of two complex conjugate binomials.

    $(x+7i)(x-7i)$. $x^2 + 49 = x^2 + (7i)^2$ factors as conjugate pair.

34. Rewrite $x^2+8$ as a product of two complex conjugate binomials.

    $(x+2i\sqrt{2})(x-2i\sqrt{2})$. $8 = 4 \cdot 2$, so $\sqrt{8} = 2\sqrt{2}$.

35. Rewrite $x^2+4$ as a product of two complex conjugate binomials.

    $(x+2i)(x-2i)$. $x^2 + 4 = x^2 + (2i)^2$ factors as conjugate pair.

36. State the Complex Conjugate Root Theorem for polynomials with real coefficients.

    If $a+bi$ is a root, then $a-bi$ is a root. Complex roots of real polynomials come in conjugate pairs.

37. Factor $x^2+4x+5$ over the complex numbers.

    $(x+2+i)(x+2-i)$. Complete the square: $(x+2)^2 - 4 + 5 = (x+2)^2 + 1$.

38. What are the complex roots of $x^2+16=0$?

    $x=4i,-4i$. Solve $x^2 = -16$ to get $x = \pm 4i$.

39. Rewrite $x^2+3$ as a product of two complex conjugate binomials.

    $(x+i\sqrt{3})(x-i\sqrt{3})$. $x^2 + 3 = x^2 + (i\sqrt{3})^2$ factors as conjugate pair.

40. Factor $x^2-4x+8$ over the complex numbers.

    $(x-2+2i)(x-2-2i)$. Complete the square: $(x-2)^2 - 4 + 8 = (x-2)^2 + 4$.

41. What are the complex roots of $x^2+2=0$?

    $x=i\sqrt{2},-i\sqrt{2}$. Solve $x^2 = -2$ to get $x = \pm i\sqrt{2}$.

42. State the identity for factoring a sum of squares using $i$.

    $a^2+b^2=(a+bi)(a-bi)$. Uses complex conjugates to factor sum of squares.

43. Factor $x^2-4x+5$ over the complex numbers.

    $(x-2+i)(x-2-i)$. Complete the square: $(x-2)^2 - 4 + 5 = (x-2)^2 + 1$.

44. What is $i^{10}$?

    $i^{10}=-1$. $i^{10} = i^{8} \cdot i^2 = 1 \cdot (-1) = -1$

45. What is the product $(a+bi)(a-bi)$ equal to, simplified?

    $a^2+b^2$. Multiplying complex conjugates eliminates the imaginary terms.

46. Rewrite $x^2+18$ as a product of two complex conjugate binomials.

    $(x+3i\sqrt{2})(x-3i\sqrt{2})$. $18 = 9 \cdot 2$, so $\sqrt{18} = 3\sqrt{2}$.

47. Rewrite $x^2+9$ as a product of two complex conjugate binomials.

    $(x+3i)(x-3i)$. $x^2 + 9 = x^2 + (3i)^2$ factors as conjugate pair.

48. Rewrite $x^2+16$ as a product of two complex conjugate binomials.

    $(x+4i)(x-4i)$. $x^2 + 16 = x^2 + (4i)^2$ factors as conjugate pair.

49. Rewrite $x^2+100$ as a product of two complex conjugate binomials.

    $(x+10i)(x-10i)$. $x^2 + 100 = x^2 + (10i)^2$ factors as conjugate pair.

50. Factor $x^2-8x+20$ over the complex numbers.

    $(x-4+2i)(x-4-2i)$. Complete the square: $(x-4)^2 - 16 + 20 = (x-4)^2 + 4$.

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