Question

Factor the following polynomial AND find the zeros (show all work): x2-5x+6

Answer 1

To factor the polynomial \( x^2 - 5x + 6 \) and find its zeros, we can follow these steps:

### Step 1: Identify the coefficients.
The polynomial is of the form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = -5 \)
- \( c = 6 \)

### Step 2: Look for two numbers that multiply to \( ac \) and add to \( b \).
We need two numbers that multiply to \( a \cdot c = 1 \cdot 6 = 6 \) and add up to \( b = -5 \).

The two numbers that satisfy these conditions are:
- \( -2 \) and \( -3 \)

### Step 3: Rewrite the middle term using these numbers.
Using these numbers, we can rewrite the polynomial:
\[
x^2 - 5x + 6 = x^2 - 2x - 3x + 6
\]

### Step 4: Factor by grouping.
Next, we can factor by grouping:
\[
= (x^2 - 2x) + (-3x + 6)
\]
\[
= x(x - 2) - 3(x - 2)
\]
\[
= (x - 2)(x - 3)
\]

### Step 5: Write the factored form.
The factored form of the polynomial is:
\[
(x - 2)(x - 3)
\]

### Step 6: Find the zeros.
To find the zeros, we set each factor equal to zero:
1. \( x - 2 = 0 \)
- This gives \( x = 2 \)

2. \( x - 3 = 0 \)
- This gives \( x = 3 \)

### Summary
The factored form of the polynomial \( x^2 - 5x + 6 \) is:
\[
(x - 2)(x - 3)
\]
The zeros of the polynomial are:
\[
x = 2 \quad \text{and} \quad x = 3
\]