Fundamental Theorem of Algebra for Quadratics - Algebra 2

1. Given roots $r_1=5$ and $r_2=-1$, what is the monic quadratic polynomial?

   $x^2-4x-5$. Using sum=$4$ and product=$-5$ in $x^2-(sum)x+(product)$.

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

   $x=i$ and $x=-i$. Solving $x^2=-1$ gives $x=\pm\sqrt{-1}=\pm i$.

3. For $x^2+px+q=0$, what are the sum and product of the roots?

   $r_1+r_2=-p$ and $r_1r_2=q$. For monic quadratic $x^2+px+q$, coefficients relate directly to roots.

4. Write the monic quadratic with roots $-4+i$ and $-4-i$.

   $x^2+8x+17$. Using Vieta's formulas: sum=$-8$, product=$17$.

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

   $(x-(1+2i))(x-(1-2i))$. Using roots $1+2i$ and $1-2i$ to form linear factors.

6. What is the relationship between complex roots and linear factors over $\mathbb{C}$ for a quadratic?

   A quadratic factors into two linear factors over $\mathbb{C}$. FTA guarantees complete factorization over complex numbers.

7. If $\Delta<0$ in $x=\frac{-b\pm\sqrt{\Delta}}{2a}$, what form do the two solutions take?

   $x=\frac{-b}{2a}\pm i\frac{\sqrt{-\Delta}}{2a}$. Standard form showing complex conjugate pair structure.

8. If $ax^2+bx+c$ has roots $r_1,r_2$, what is $c$ in terms of $a$ and $r_1r_2$?

   $c=a(r_1r_2)$. From Vieta's formulas, product of roots equals $\frac{c}{a}$.

9. Write $x^2+6x+9$ in factored form showing multiplicity.

   $(x+3)^2$. Perfect square trinomial factors as $(x+3)^2$.

10. Which discriminant condition gives two distinct real roots for $ax^2+bx+c=0$?

    $\Delta>0$. Positive discriminant means the square root yields two real values.

11. What is the degree of a quadratic polynomial written as $ax^2+bx+c$ with $a\ne 0$?

    Degree $2$. The highest power term $ax^2$ determines the degree is $2$.

12. Which discriminant condition gives exactly one real root (a repeated root) for $ax^2+bx+c=0$?

    $\Delta=0$. Zero discriminant makes the $\pm$ term disappear, giving one solution.

13. Find the complex roots of $x^2-2x+2=0$.

    $x=1\pm i$. Using quadratic formula with $\Delta=-4<0$.

14. Which discriminant condition gives two nonreal complex roots for $ax^2+bx+c=0$?

    $\Delta<0$. Negative discriminant requires imaginary unit for square root.

15. What is the repeated root of $x^2+6x+9=0$?

    $x=-3$. Since $\Delta=0$, there's one repeated root.

16. What does it mean for a root of a polynomial to have multiplicity $2$?

    The factor $(x-r)^2$ divides the polynomial. Multiplicity $2$ means the root appears twice in factorization.

17. What is the standard factorization form of a quadratic with roots $r_1$ and $r_2$?

    $a(x-r_1)(x-r_2)$. Each root contributes a linear factor $(x-r_i)$.

18. Find the complex roots of $x^2+10x+34=0$.

    $x=-5\pm 3i$. Using quadratic formula with $\Delta=-36<0$.

19. Find the discriminant $\Delta$ of $x^2+6x+9=0$.

    $\Delta=0$. Using $\Delta=b^2-4ac$ with $a=1,b=6,c=9$.

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

    $(x-(-1+2i))(x-(-1-2i))$. Using roots $-1+2i$ and $-1-2i$ to form linear factors.

21. Find the roots of $7x^2-1=0$.

    $x=\pm\frac{1}{\sqrt{7}}$. Solving $7x^2=1$ gives $x^2=\frac{1}{7}$.

22. How many real roots does $x^2-4x+7=0$ have, based on $\Delta$?

    $0$ real roots. Since $\Delta<0$, there are no real roots.

23. If $ax^2+bx+c$ has roots $r_1,r_2$, what is $b$ in terms of $a$ and $r_1+r_2$?

    $b=-a(r_1+r_2)$. From Vieta's formulas, sum of roots equals $-\frac{b}{a}$.

24. Find $r_1+r_2$ for $3x^2-12x+1=0$ without solving.

    $r_1+r_2=4$. Sum formula: $r_1+r_2=-\frac{b}{a}=-\frac{-12}{3}=4$.

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

    $(x-(a+bi))(x-(a-bi))$. Linear factors corresponding to conjugate pair roots.

26. What does the Complex Conjugate Root Theorem say for polynomials with real coefficients?

    If $a+bi$ is a root, then $a-bi$ is a root. Real coefficients force complex roots to come in conjugate pairs.

27. Find the roots of $4x^2+4x+5=0$ in $a\pm bi$ form.

    $x=-\frac{1}{2}\pm i$. Using quadratic formula with $\Delta=-64<0$.

28. Find the complex solutions of $x^2-4x+7=0$ in $a\pm bi$ form.

    $x=2\pm i\sqrt{3}$. Using quadratic formula with $\Delta=-12<0$.

29. Identify the complex roots of $x^2+9=0$.

    $x=3i$ and $x=-3i$. Solving $x^2=-9$ gives $x=\pm\sqrt{-9}=\pm 3i$.

30. Find the solutions of $2x^2+3x-2=0$.

    $x=\frac{1}{2}$ and $x=-2$. Using quadratic formula with $\Delta=25>0$.

31. What does it mean to count roots "with multiplicity" for a quadratic?

    A double root counts as $2$ roots total. Repeated roots are counted multiple times in the total.

32. Find $r_1r_2$ for $3x^2-12x+1=0$ without solving.

    $r_1r_2=\frac{1}{3}$. Product formula: $r_1r_2=\frac{c}{a}=\frac{1}{3}$.

33. What is the quadratic formula for the solutions of $ax^2+bx+c=0$?

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Standard formula derived from completing the square method.

34. What does the Fundamental Theorem of Algebra say about the number of complex roots of a degree $n$ polynomial (counting multiplicity)?

    Exactly $n$ complex roots, counting multiplicity. FTA states degree $n$ polynomial has exactly $n$ complex roots when counting multiplicity.

35. Identify the missing statement to show FTA for quadratics: $\Delta<0\Rightarrow\sqrt{\Delta}=i\sqrt{-\Delta}$.

    $\sqrt{\Delta}=i\sqrt{-\Delta}$. When discriminant is negative, square root becomes imaginary.

36. Find the complex roots of $x^2-8x+20=0$.

    $x=4\pm 2i$. Using quadratic formula with $\Delta=-16<0$.

37. What does the Fundamental Theorem of Algebra guarantee for any nonconstant polynomial with complex coefficients?

    It has at least one complex root. FTA guarantees at least one complex root for any nonconstant polynomial.

38. What is the definition of the imaginary unit $i$?

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

39. How do you rewrite $\sqrt{-k}$ for $k>0$ using $i$?

    $\sqrt{-k}=i\sqrt{k}$. Standard method to express square roots of negative numbers.

40. Identify the conjugate of the complex root $3-5i$.

    $3+5i$. Complex conjugate of $3-5i$ flips the imaginary sign.

41. What is the conjugate of a complex number $a+bi$?

    $a-bi$. Complex conjugate changes the sign of the imaginary part.

42. What are the sum and product of roots for $ax^2+bx+c=0$ in terms of $a,b,c$?

    $r_1+r_2=-\frac{b}{a}$ and $r_1r_2=\frac{c}{a}$. Vieta's formulas relate coefficients to root combinations.

43. If a real-coefficient quadratic has root $-1+2i$, what is the other root?

    $-1-2i$. Complex Conjugate Root Theorem for real coefficients.

44. Find the discriminant $\Delta$ of $2x^2+3x-2=0$.

    $\Delta=25$. Using $\Delta=b^2-4ac$ with $a=2,b=3,c=-2$.

45. What does the Fundamental Theorem of Algebra imply about the total number of complex zeros of a quadratic (counting multiplicity)?

    It has $2$ complex zeros, counting multiplicity. Since degree is $2$, FTA guarantees exactly $2$ complex zeros.

46. Identify the number of complex roots (counting multiplicity) of $p(x)=7x^2-1$.

    $2$. Degree $2$ polynomial always has exactly $2$ complex roots.

47. What does the Factor Theorem state about $r$ and the factor $(x-r)$?

    $p(r)=0\iff(x-r)$ is a factor of $p(x)$. Fundamental connection between roots and linear factors.

48. What is the discriminant of $ax^2+bx+c$?

    $\Delta=b^2-4ac$. The discriminant determines the nature of the roots.

49. For $x^2-2ax+(a^2+b^2)=0$, what are the roots in terms of $a$ and $b$?

    $x=a\pm bi$. Standard form for conjugate pair roots $a\pm bi$.

50. Find the roots of $x^2-6x+13=0$ in $a\pm bi$ form.

    $x=3\pm 2i$. Using quadratic formula with discriminant $\Delta=-16<0$.

51. Find the discriminant $\Delta$ of $x^2-4x+7=0$.

    $\Delta=-12$. Using $\Delta=b^2-4ac$ with $a=1,b=-4,c=7$.

52. Write the monic quadratic with roots $2+3i$ and $2-3i$.

    $x^2-4x+13$. Using Vieta's formulas: sum=$4$, product=$13$.

53. Find the complex roots of $x^2+2x+2=0$.

    $x=-1\pm i$. Using quadratic formula with $\Delta=-4<0$.

54. Find the complex solutions of $x^2+4x+8=0$ in $a\pm bi$ form.

    $x=-2\pm 2i$. Using quadratic formula with negative discriminant.

55. What is the key conclusion that verifies the Fundamental Theorem of Algebra for quadratics?

    Every quadratic has $2$ complex roots, counting multiplicity. This confirms FTA applies to all degree $2$ polynomials.

56. What does it mean for a complex number to be a root of $p(x)$?

    $p(r)=0$. A root makes the polynomial evaluate to zero.

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