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’ll follow these steps:

### Step 1: Factor the Polynomial

We are looking for two numbers that:
1. Multiply to give the constant term (which is \( 6 \)).
2. Add up to give the coefficient of the linear term (which is \( -5 \)).

We need two numbers, let’s say \( a \) and \( b \), such that:
- \( a \cdot b = 6 \)
- \( a + b = -5 \)

The pairs of factors of \( 6 \) are:
- \( (1, 6) \)
- \( (2, 3) \)
- \( (-1, -6) \)
- \( (-2, -3) \)

Now we can check which of these pairs adds up to \( -5 \):
- \( -2 + (-3) = -5 \)

Thus, we can factor the polynomial as:
\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\]

### Step 2: Find the Zeros

To find the zeros of the polynomial, we set the factored expression equal to zero:
\[
(x - 2)(x - 3) = 0
\]

Now we can set each factor to zero:
1. \( x - 2 = 0 \) which gives \( x = 2 \)
2. \( x - 3 = 0 \) which gives \( x = 3 \)

### Conclusion

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