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:
- Multiply to give the constant term (which is \( 6 \)).
- 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:
- \( x - 2 = 0 \) which gives \( x = 2 \)
- \( x - 3 = 0 \) 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: \[ x = 2 \quad \text{and} \quad x = 3 \]