Factor the trinomial completely. \[ p^{2}-12 p+35 \]

To factor the trinomial \( p^{2} - 12p + 35 \) completely, follow these steps: ### Step 1: Identify the Coefficients The given quadratic trinomial is in the form: \[ p^{2} + bp + c \] where: - \( b = -12 \) - \( c = 35 \) ### Step 2: Find Two Numbers That Multiply to \( c \) and Add to \( b \) We need to find two numbers, let's call them \( m \) and \( n \), such that: \[ m \times n = c = 35 \] \[ m + n = b = -12 \] Let's list the pairs of factors of 35: - \( 1 \) and \( 35 \) - \( 5 \) and \( 7 \) Since the constant term \( c \) is positive and the middle term \( b \) is negative, both numbers \( m \) and \( n \) must be negative. Check the pair \( -5 \) and \( -7 \): - \( -5 \times -7 = 35 \) ✅ - \( -5 + (-7) = -12 \) ✅ ### Step 3: Write the Factored Form Using the numbers \( -5 \) and \( -7 \), the trinomial can be factored as: \[ p^{2} - 12p + 35 = (p - 5)(p - 7) \] ### Verification To ensure the factorization is correct, expand the factored form: \[ (p - 5)(p - 7) = p \times p + p \times (-7) + (-5) \times p + (-5) \times (-7) \\ = p^{2} - 7p - 5p + 35 \\ = p^{2} - 12p + 35 \] The expanded form matches the original trinomial, confirming that the factorization is correct. ### Final Answer \[ p^{2} - 12p + 35 = (p - 5)(p - 7) \]