Multiple choice

The number of imaginar roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=32$ is

  1. $Zero$
  2. $1$
  3. $2$
  4. $4$
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A Correct answer
AI explanation

To solve the equation, we first rearrange the factors on the left side. We pair the first and fourth factors, and the second and third factors: (x - 1)(3x + 1) = 3x squared - 2x - 1, and (x - 2)(3x - 2) = 3x squared - 8x + 4. Let y = 3x squared - 5x. Adding and subtracting 3x in the first expression gives (y - 3x) - 1, and adding and subtracting 3x in the second gives (y + 3x) + 4. The equation becomes (y - 3x - 1)(y + 3x + 4) = 32. Expanding this yields (y - 1)(y + 4) + 3x(y - 1) - 3x(y + 4) - 12x squared = 32. Wait, expanding properly: (y - (3x + 1))(y + (3x + 4)) = y squared + y(3x + 4 - 3x - 1) - (3x + 1)(3x + 4) = y squared + 3y - (9x squared + 15x + 4) = 32. Since y = 3x squared - 5x, we have 9x squared + 15x = 3(3x squared + 5x) = 3(3x squared - 5x + 10x) = 3y + 30x. This path seems complicated, let's try substituting y = 3x squared - 5x directly into the original factors. The first factor is y - 3x - 1 and the second is y + 3x + 4. Wait, (y - 3x - 1)(y + 3x + 4) = y squared + 3y - (3x + 1)(3x + 4) = 32. Notice that 9x squared + 15x + 4 = 3(3x squared + 5x) + 4. Let y = 3x squared - 5x. Then (x - 1)(3x + 1) = 3x squared - 2x - 1 = y + 3x - 1. And (x - 2)(3x - 2) = 3x squared - 8x + 4 = y - 3x + 4. The product is (y + 3x - 1)(y - 3x + 4) = 32. Expanding this gives y squared + 3xy + 4y - 3xy - 9x squared + 12x - y - 3x + 4 = y squared + 3y - 9x squared + 9x + 4 = 32. Substituting y = 3x squared - 5x gives 9x squared - 30x + 3x squared + 9x squared + 9x - 9x squared - 9x + 4 = 32. This yields 12x squared - 30x + 4 = 32, or 6x squared - 15x - 14 = 0. The discriminant is b squared - 4ac = 225 - 4(6)(-14) = 225 + 336 = 561. Since the discriminant is positive, there are two distinct real roots. A polynomial equation of degree 4 can have at most 4 roots. To find the number of imaginary roots, we can graph the function or use calculus to show that it only crosses the x-axis twice. The number of imaginary roots is Zero.

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