Find the degree of the polynomial `(x+2)(x^(2)-2x+4)` .

Answer

To find the degree of the polynomial \((x + 2)(x^2 - 2x + 4)\), we will follow these steps:

### Step 1: Expand the polynomial
We start by expanding the expression \((x + 2)(x^2 - 2x + 4)\).

Using the distributive property (also known as the FOIL method for binomials), we multiply each term in the first polynomial by each term in the second polynomial:

\[ (x + 2)(x^2 - 2x + 4) = x \cdot (x^2 - 2x + 4) + 2 \cdot (x^2 - 2x + 4) \]

### Step 2: Distribute the terms
Now, we will distribute \(x\) and \(2\):

1. For \(x \cdot (x^2 - 2x + 4)\):
- \(x \cdot x^2 = x^3\)
- \(x \cdot (-2x) = -2x^2\)
- \(x \cdot 4 = 4x\)

So, \(x \cdot (x^2 - 2x + 4) = x^3 - 2x^2 + 4x\).

2. For \(2 \cdot (x^2 - 2x + 4)\):
- \(2 \cdot x^2 = 2x^2\)
- \(2 \cdot (-2x) = -4x\)
- \(2 \cdot 4 = 8\)

So, \(2 \cdot (x^2 - 2x + 4) = 2x^2 - 4x + 8\).

### Step 3: Combine like terms
Now, we will combine the results from both distributions:

\[ (x^3 - 2x^2 + 4x) + (2x^2 - 4x + 8) \]

Combining like terms:
- The \(x^3\) term: \(x^3\)
- The \(x^2\) terms: \(-2x^2 + 2x^2 = 0\)
- The \(x\) terms: \(4x - 4x = 0\)
- The constant term: \(8\)

Thus, the polynomial simplifies to:

\[ x^3 + 8 \]

### Step 4: Determine the degree
The degree of a polynomial is defined as the highest power of \(x\) in the polynomial. In our case, the highest power of \(x\) is \(3\).

Therefore, the degree of the polynomial \((x + 2)(x^2 - 2x + 4)\) is:

\[ \text{Degree} = 3 \]

### Summary
The degree of the polynomial \((x + 2)(x^2 - 2x + 4)\) is \(3\).

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