Polynomial Identities and Complex Numbers (Grades 11-12)

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Polynomial Identities and Complex Numbers

1. 
Factor. [math]x^2+16[/math]
  1. [math]sqrt16i[/math]
  2. [math](x-4i)(x+4i)[/math]
  3. [math]+-sqrt4[/math]
  4. [math](x+2i)(x-2i)[/math]
2. 
Factor. [math]x^2+12[/math]
  1. [math]2sqrt3i[/math]
  2. [math](x-4i)(x+4i)[/math]
  3. [math]+-sqrt12[/math]
  4. [math](x+2sqrt3i)(x-2sqrt3i)[/math]
3. 
Factor. [math]x^2+64[/math]
  1. [math](x+8i)(x-8i)[/math]
  2. [math](x-8)(x+8)[/math]
  3. [math]+-sqrt64[/math]
  4. [math](x+64i)(x-64i)[/math]
4. 
Which of the following is the factored form of [math]x^3 + 8 ?[/math]
  1. [math](x+2)(x-1+sqrt(3) \ i)(x-1-sqrt(3) \ i)[/math]
  2. [math](x+2)(x-2)(x+2)[/math]
  3. [math](x+2)(x-2i)(x+2i)[/math]
  4. [math](x+2)(x+2+2sqrt(3) \ i)(x-2-2sqrt(3) \ i)[/math]
5. 
Factor [math]x^4-1[/math].
  1. [math](x^2+1)(x+1)(x+i)[/math]
  2. [math](x+1)(x-1)(x+1+i)(x-1-i)[/math]
  3. [math](x+1)(x-1)(x+i)(x-i)[/math]
  4. [math](x+1)(x-1)(x+sqrt(2)i)(x-sqrt(2)i)[/math]
6. 
If [math]x - 3/2 + 3/2sqrt(3) \ i[/math] is a factor of [math]x^3 + 27[/math], which of the following are also factors of this polynomial?
  1. [math]x - 3[/math]
  2. [math]x + 3/2 - 3/2 sqrt(3)[/math]
  3. [math]x - 3/2 - 3/2 sqrt(3) \ i [/math]
  4. [math]x + 3/2 - 3/2 sqrt(3) \ i[/math]
7. 
If [math]x+2(1-sqrt(3)i)[/math] is a factor of [math]x^3-64[/math], which of the following are also factors? Choose all that apply.
  1. [math]x-4[/math]
  2. [math]x+2(1+sqrt(3)i)[/math]
  3. [math]x-2(1-sqrt(3)i)[/math]
  4. [math]x-2(1+sqrt(3)i)[/math]
8. 
Which of the following are equivalent to [math]x^4 - 16 ?[/math] Choose all that apply.
  1. [math](x+2)(x^3-2i \ x^2 - 4x + 8i)[/math]
  2. [math](x+2)(x-2)(x+2i)(x-2i)[/math]
  3. [math](x+2)(x-2)(x^2+4)[/math]
  4. [math](x+2i)(x-2i)(x^2-4)[/math]
9. 
Which of the following are equivalent to [math]4x^4 - 81 ?[/math] Choose all that apply.
  1. [math](sqrt(2)x - 3)(sqrt(2)x + 3)(sqrt(2)x - 3i)(sqrt(2)x + 3i)[/math]
  2. [math](root[4](4)x - 3)^2(root[4](4)x + 3)^2[/math]
  3. [math](2x^2 - 9)(2x^2 + 9)[/math]
  4. [math](1/2 x + 3)(1/2 x -3) (2x^2 + 9)[/math]
10. 
For the polynomial [math] P(x) = (ax)^3 + b^3[/math], [math]a,b in RR[/math], and where [math]a,b[/math] are positive, if [math]1 + sqrt(3)i [/math] is a root, and in fully factored form there is a leading constant of 8, what are the coefficients [math]a,b ?[/math] (Assume that the coefficient of 8 has come from factoring out a 2 from each of the three factors of [math]P(x)[/math]).
  1. [math]a=1, b=2[/math]
  2. [math]a=1, b=4[/math]
  3. [math]a=2, b=4[/math]
  4. Not enough information to solve.

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