Find a polynomial function with real coefficients that has the given numbers as roots: 4, 0, 3, italic i
a)x^4-4x^3 +9x^2-36x
b)x^3-4x^2 -3x+12
c)x^3-4x^2 +3x-12
d)4x-4x^3 -9x^2 +36x
My gut instinct is either b or c but I could be wrong.
If it were b or c, x=0 would not work.
a and d do not work with x = i.
None of those choices have all of those four solutions. A polynomial that works is
x(x^2+1)(x-3)(x-4)
To find a polynomial function with real coefficients that has the given numbers as roots (4, 0, 3, and i), you need to consider each option and check if all the roots are satisfied.
Let's go through each option and check if it satisfies all the given roots:
a) x^4 - 4x^3 + 9x^2 - 36x
Checking the roots:
- For x = 4: (4)^4 - 4(4)^3 + 9(4)^2 - 36(4) = 256 - 512 + 576 - 144 = 0 ✓
- For x = 0: (0)^4 - 4(0)^3 + 9(0)^2 - 36(0) = 0 - 0 + 0 - 0 = 0 ✓
- For x = 3: (3)^4 - 4(3)^3 + 9(3)^2 - 36(3) = 81 - 108 + 81 - 108 = -54 ≠ 0 ✗
- For x = i: (i)^4 - 4(i)^3 + 9(i)^2 - 36(i) = 1 - 4i - 9 - 36i = -8 - 40i ≠ 0 ✗
b) x^3 - 4x^2 - 3x + 12
Checking the roots:
- For x = 4: (4)^3 - 4(4)^2 - 3(4) + 12 = 64 - 64 - 12 + 12 = 0 ✓
- For x = 0: (0)^3 - 4(0)^2 - 3(0) + 12 = 12 ≠ 0 ✗
- For x = 3: (3)^3 - 4(3)^2 - 3(3) + 12 = 27 - 36 - 9 + 12 = -6 ≠ 0 ✗
- For x = i: (i)^3 - 4(i)^2 - 3(i) + 12 = -i - 4(-1) - 3i + 12 = 11 - 4i ≠ 0 ✗
c) x^3 - 4x^2 + 3x - 12
Checking the roots:
- For x = 4: (4)^3 - 4(4)^2 + 3(4) - 12 = 64 - 64 + 12 - 12 = 0 ✓
- For x = 0: (0)^3 - 4(0)^2 + 3(0) - 12 = -12 ≠ 0 ✗
- For x = 3: (3)^3 - 4(3)^2 + 3(3) - 12 = 27 - 36 + 9 - 12 = -12 ≠ 0 ✗
- For x = i: (i)^3 - 4(i)^2 + 3(i) - 12 = -i - 4(-1) + 3i - 12 = -11 - 2i ≠ 0 ✗
d) 4x - 4x^3 - 9x^2 + 36x
Checking the roots:
- For x = 4: 4(4) - 4(4)^3 - 9(4)^2 + 36(4) = 16 - 256 - 576 + 144 = -672 ≠ 0 ✗
- For x = 0: 4(0) - 4(0)^3 - 9(0)^2 + 36(0) = 0 - 0 - 0 + 0 = 0 ✓
- For x = 3: 4(3) - 4(3)^3 - 9(3)^2 + 36(3) = 12 - 108 - 243 + 108 = -231 ≠ 0 ✗
- For x = i: 4(i) - 4(i)^3 - 9(i)^2 + 36(i) = 4i - 4i^3 - 9(-1) + 36i = 4i + 4i - 9 + 36i = 44i - 9 ≠ 0 ✗
None of the options (a, b, c, and d) satisfy all the given roots. However, we can find a polynomial that does have these roots by multiplying the factors corresponding to each root:
Polynomial Function: x(x^2 + 1)(x - 3)(x - 4)
This polynomial has real coefficients and has the given numbers as roots: 4, 0, 3, and i.