Find the zeros of the polynomial ` x^(2) + 2x - 195.`

Answer

To find the zeros of the polynomial \( x^2 + 2x - 195 \), we will follow these steps:

### Step 1: Set the polynomial equal to zero
We start with the polynomial:
\[ f(x) = x^2 + 2x - 195 \]
To find the zeros, we set \( f(x) = 0 \):
\[ x^2 + 2x - 195 = 0 \]

### Step 2: Factor the quadratic polynomial
We need to factor the quadratic expression. We look for two numbers that multiply to \(-195\) (the constant term) and add up to \(2\) (the coefficient of \(x\)).

The two numbers that satisfy this condition are \(15\) and \(-13\) because:
\[ 15 \times (-13) = -195 \quad \text{and} \quad 15 + (-13) = 2 \]

### Step 3: Rewrite the polynomial using the factors
Using the numbers we found, we can rewrite the polynomial:
\[ x^2 + 15x - 13x - 195 = 0 \]

### Step 4: Group the terms
Now, we can group the terms:
\[ (x^2 + 15x) + (-13x - 195) = 0 \]

### Step 5: Factor by grouping
Next, we factor out the common terms from each group:
\[ x(x + 15) - 13(x + 15) = 0 \]
Now, we can factor out \( (x + 15) \):
\[ (x + 15)(x - 13) = 0 \]

### Step 6: Set each factor to zero
Now, we set each factor equal to zero:
1. \( x + 15 = 0 \)
2. \( x - 13 = 0 \)

### Step 7: Solve for \(x\)
From the first equation:
\[ x + 15 = 0 \implies x = -15 \]
From the second equation:
\[ x - 13 = 0 \implies x = 13 \]

### Conclusion
The zeros of the polynomial \( x^2 + 2x - 195 \) are:
\[ x = -15 \quad \text{and} \quad x = 13 \]

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