Question

Factor the following polynomial AND find the zeros (show all work): x^2-5x+6

Answer 1

To factor the polynomial \(x^2 - 5x + 6\), we can look for two numbers that multiply to the constant term (6) and add up to the coefficient of the linear term (-5).

1. **Identify the relationship**:
We need two numbers \(a\) and \(b\) such that:
- \(a \cdot b = 6\) (the constant term)
- \(a + b = -5\) (the coefficient of \(x\))

2. **Find suitable pair of numbers**:
The pairs of factors of 6 are:
- \(1\) and \(6\)
- \(2\) and \(3\)
- \(-1\) and \(-6\)
- \(-2\) and \(-3\)

From these pairs, we see that \(-2\) and \(-3\) satisfy both conditions:
- \(-2 \cdot -3 = 6\)
- \(-2 + -3 = -5\)

3. **Write the factored form**:
Therefore, we can factor the polynomial as:
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]

4. **Finding the zeros**:
To find the zeros, we set each factor equal to zero:
\[
x - 2 = 0 \quad \text{or} \quad x - 3 = 0
\]
Solving these equations, we find:
\[
x = 2 \quad \text{and} \quad x = 3
\]

**Final Result**:
- The factored form of the polynomial \(x^2 - 5x + 6\) is \((x - 2)(x - 3)\).
- The zeros of the polynomial are \(x = 2\) and \(x = 3\).