5) \( n^{2}-10 n+9 \)

It appears you would like to factor the quadratic expression \( n^{2} - 10n + 9 \). Let's proceed with factoring it. ### Given: \[ n^{2} - 10n + 9 \] ### Step 1: Identify the Coefficients For a quadratic expression of the form \( ax^{2} + bx + c \), identify: - \( a = 1 \) (coefficient of \( n^{2} \)) - \( b = -10 \) (coefficient of \( n \)) - \( c = 9 \) (constant term) ### Step 2: Find Two Numbers That Multiply to \( a \times c \) and Add to \( b \) We need two numbers that multiply to \( 1 \times 9 = 9 \) and add up to \( -10 \). - The numbers are **-1** and **-9** because: - \( (-1) \times (-9) = 9 \) - \( (-1) + (-9) = -10 \) ### Step 3: Rewrite the Middle Term Using These Numbers \[ n^{2} - 1n - 9n + 9 \] ### Step 4: Factor by Grouping Group the terms: \[ (n^{2} - 1n) + (-9n + 9) \] Factor out the greatest common factor (GCF) from each group: \[ n(n - 1) - 9(n - 1) \] ### Step 5: Factor Out the Common Binomial \[ (n - 1)(n - 9) \] ### Final Factored Form: \[ n^{2} - 10n + 9 = (n - 1)(n - 9) \] ### Verification: To ensure the factorization is correct, you can expand the factors: \[ (n - 1)(n - 9) = n \cdot n - 9n - 1n + 9 = n^{2} - 10n + 9 \] This matches the original expression. ### Conclusion: The quadratic expression \( n^{2} - 10n + 9 \) factors to: \[ (n - 1)(n - 9) \]