https://file.aiquickdraw.com/mathbot/file/1727365658074q1st0tqk.jpg

The problem presented in the image is in Indonesian and can be translated as follows:

**Problem 16:**
One of the real roots of the equation:
\[
x^3 + 2x^2 + ax = 0
\]
is \( x = 2 \). The other real roots are:

Options:
- A. \( 0 \) and \( -2 \)
- B. \( -1 \) and \( 1 \)
- C. \( 0 \) and \( -1 \)
- D. \( 0 \) and \( 1 \)
- E. None

### Solution Approach:

We are given one of the roots, \( x = 2 \). Using this, we can factor the polynomial by dividing it by \( (x - 2) \).

The equation is:
\[
x^3 + 2x^2 + ax = 0
\]
This can be factored as:
\[
x(x^2 + 2x + a) = 0
\]
One of the roots is \( x = 2 \), so we can divide \( x^2 + 2x + a \) by \( (x - 2) \) to find the remaining quadratic.

I'll now compute the remaining roots using this information.The division of the polynomial \( x^3 + 2x^2 + ax \) by \( (x - 2) \) yields a quotient of:

\[
x^2 + 4x + (a + 8)
\]

and a remainder of \( 2a + 16 \). Since \( x = 2 \) is a root, the remainder must be zero:

\[
2a + 16 = 0
\]

Solving for \( a \):

\[
a = -8
\]

Thus, the polynomial becomes:

\[
x^3 + 2x^2 - 8x = 0
\]
or
\[
x(x^2 + 2x - 8) = 0
\]

Now, we can solve \( x^2 + 2x - 8 = 0 \) to find the remaining roots. I'll compute that next.The quadratic equation \( x^2 + 2x - 8 = 0 \) has the roots \( x = -4 \) and \( x = 2 \). 

Therefore, the complete set of real roots for the equation \( x^3 + 2x^2 - 8x = 0 \) are:
\[
x = 0, x = 2, x = -4
\]

Since one of the roots is already given as \( x = 2 \), the other real roots are \( x = 0 \) and \( x = -4 \).

Thus, none of the answer choices in the question directly match these values. Hence, the correct answer is:
\[
\boxed{\text{E. Tidak ada (None)}}
\]

Would you like more details on any part of the solution, or do you have any further questions?

### Related Questions:
1. How do you perform polynomial division with synthetic division?
2. Why does the remainder need to be zero when using a root to factor a polynomial?
3. What are the general methods for solving cubic equations?
4. How can you graphically interpret the roots of a polynomial equation?
5. What happens if a cubic polynomial has complex roots?

**Tip:** When solving polynomials, once you know one root, you can factor the polynomial to reduce the degree and find the remaining roots more easily.

Salah satu akar real dari persamaan x^3 + 2x^2 + ax = 0 adalah x = 2. Akar real lainnya adalah?

Learn how to solve the cubic equation x^3 + 2x^2 + ax = 0 given that x = 2 is a root. This problem explores factoring cubic equations and finding real roots using algebraic methods.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation