Given the cubic equation $x^3 + ax^2 + 25x + b = 0$ where $2 + 3i$ is one of the roots, we are tasked with finding the values of the constants $a$ and $b$ and the other roots of the equation.

Step 1: Use the complex conjugate root theorem

Since the coefficients of the polynomial are real, the complex conjugate of $2 + 3i$, which is $2 - 3i$, must also be a root.

Step 2: Express the cubic polynomial

If $2 + 3i$ and $2 - 3i$ are roots, then we can express the cubic polynomial as: $(x - (2 + 3i))(x - (2 - 3i))(x - r) = 0$ where $r$ is the remaining real root.

Step 3: Expand the product of the conjugate pair

First, compute the product of the two conjugate factors: $(x - 2 - 3i)(x - 2 + 3i) = \left(x - 2\right)^2 - (3i)^2 = (x - 2)^2 + 9$ $= x^2 - 4x + 4 + 9 = x^2 - 4x + 13$

Step 4: Multiply by the remaining root

Now, multiply by $(x - r)$: $(x^2 - 4x + 13)(x - r) = x^3 - rx^2 - 4x^2 + 4rx + 13x - 13r$ $= x^3 + (-r - 4)x^2 + (4r + 13)x - 13r$

Step 5: Match coefficients with the original equation

Now, compare this with the original equation $x^3 + ax^2 + 25x + b = 0$. This gives us: $a = -(r + 4), \quad 25 = 4r + 13, \quad b = -13r$

Step 6: Solve for $r$

From $25 = 4r + 13$: $4r = 12 \quad \Rightarrow \quad r = 3$

Step 7: Determine $a$ and $b$

Substitute $r = 3$ into the equations for $a$ and $b$: $a = -(3 + 4) = -7$ $b = -13 \times 3 = -39$

Step 8: Write down the other root

Given that $r = 3$, the other root besides $2 + 3i$ and $2 - 3i$ is $x = 3$.

Conclusion

• The value of $a$ is $-7$.
• The value of $b$ is $-39$.
• The other roots are $2 - 3i$ and $3$.

Thus, the values are:

• $a = -7$
• $b = -39$
• The remaining roots are $x = 3$ and $x = 2 - 3i$.

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Do you want more details or have any questions? Here are some related questions:

1. How do we know the conjugate of a complex root is also a root?
2. What happens if we miscalculate the roots of a polynomial?
3. How do you determine the nature of the roots based on the discriminant of a cubic equation?
4. What is the significance of the complex conjugate root theorem?
5. How can you verify the roots by substituting them back into the original equation?

Tip: Always double-check calculations when dealing with complex numbers to avoid errors in the real and imaginary parts.