f(x)=x 
5
 +6x 
4
 +15x 
3
 +30x 
2
 +44x+24 solve showing all steps

To solve the polynomial \( f(x) = x^5 + 6x^4 + 15x^3 + 30x^2 + 44x + 24 \), we'll look for the roots (i.e., the values of \( x \) where \( f(x) = 0 \)).

### Step 1: Factorization Attempt

1. **Use the Rational Root Theorem**: The possible rational roots of \( f(x) \) can be found by evaluating the factors of the constant term (24) divided by the factors of the leading coefficient (1). The factors of 24 are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 \).

2. **Testing Possible Roots**:
   - Start with \( x = -2 \):
   \[
   f(-2) = (-2)^5 + 6(-2)^4 + 15(-2)^3 + 30(-2)^2 + 44(-2) + 24
   \]
   \[
   = -32 + 96 - 120 + 120 - 88 + 24 = 0
   \]
   Thus, \( x = -2 \) is a root.

### Step 2: Polynomial Division

Now that we know \( x + 2 \) is a factor, we can perform synthetic division or polynomial long division on \( f(x) \) by \( x + 2 \).

**Using Synthetic Division:**
- Coefficients of \( f(x) \): \( 1, 6, 15, 30, 44, 24 \)

```
-2 | 1  6  15  30  44  24
   |    -2   8  14  -8  -24
-------------------------
     1  4  7  16  20   0
```

The result is \( x^4 + 4x^3 + 7x^2 + 16x + 20 \).

### Step 3: Solve the Quartic Polynomial

Now we need to factor \( x^4 + 4x^3 + 7x^2 + 16x + 20 \). We'll use the Rational Root Theorem again.

#### Testing Roots:
- Testing \( x = -2 \):
\[
f(-2) = (-2)^4 + 4(-2)^3 + 7(-2)^2 + 16(-2) + 20
\]
\[
= 16 - 32 + 28 - 32 + 20 = 0
\]
So \( x + 2 \) is also a factor of this quartic polynomial.

**Synthetic Division Again**:
Using \( x + 2 \):

```
-2 | 1  4  7  16  20
   |   -2  -4  -6  -8
----------------------
     1  2  3  10   0
```

Now we have:
\[
x^3 + 2x^2 + 3x + 10
\]

### Step 4: Solve the Cubic Polynomial

Now we need to find the roots of \( x^3 + 2x^2 + 3x + 10 \). 

#### Testing Roots:
None of the rational roots \( \pm 1, \pm 2, \pm 5, \pm 10 \) work. We can use numerical methods or the cubic formula, but let's try finding one real root by trial or using synthetic division with a guessed value.

### Step 5: Use Numerical Methods or Cubic Formula

Assuming we can't find a simple rational root, we might check:
- **Numerical Method** (e.g., Newton's method) or graphing to estimate roots, or the cubic formula for a general solution.

### Summary of Roots

1. We found two factors: \( x + 2 \) twice.
2. The cubic \( x^3 + 2x^2 + 3x + 10 \) needs further analysis for roots.

### Conclusion

The factors found are:
\[
f(x) = (x + 2)^2 \cdot (x^3 + 2x^2 + 3x + 10)
\]
To find exact roots of the cubic, numerical methods may be applied or it can be graphically analyzed.

If you need more detailed steps for the cubic part or further analysis, let me know!

### Additional Questions
1. Would you like to see the steps for using the cubic formula?
2. Do you want to explore numerical methods for root finding?
3. Are you interested in graphing the polynomial to visualize the roots?
4. Would you like more details on synthetic division?
5. Do you have a specific question about polynomial functions?

### Tip
When solving higher-degree polynomials, always check for simple rational roots first to simplify the problem!

f(x)=x^5 + 6x^4 + 15x^3 + 30x^2 + 44x + 24 solve showing all steps

Explore the detailed solution for the polynomial f(x)=x^5 + 6x^4 + 15x^3 + 30x^2 + 44x + 24. This guide covers methods such as synthetic division and the Rational Root Theorem, suitable for high school students in Grades 10-12.

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