Ask AI: factor 15n²-27n-6

To factor the quadratic expression \(15n² - 27n - 6\), we need to find two binomials whose product gives the original quadratic expression. We start with polynomial factorization techniques. First, recognize the quadratic in its general form: \(Ax^2 + Bx + C\). For \(15n² - 27n - 6\): 1. **Identify A, B, and C:** - A (coefficient of \(n²\)) = 15 - B (coefficient of \(n\)) = -27 - C (constant term) = -6 2. **Find two numbers that multiply to \(A \cdot C\) and add to \(B\):** - \(A \cdot C = 15 \times -6 = -90\) - We need to find two numbers that multiply to -90 and add to -27. After checking possible pairs, we find that -30 and +3 work: - (-30) + 3 = -27 - (-30) * 3 = -90 3. **Rewrite the middle term using the found pair:** \(15n² - 27n - 6\) can be rewritten as \(15n² - 30n + 3n - 6\). 4. **Factor by grouping:** Group the terms to factor out the common factors from each group: - \(15n² - 30n + 3n - 6\) - Group into two pairs: \((15n² - 30n) + (3n - 6)\) 5. **Factor out the greatest common factor from each pair:** - \(15n(n - 2) + 3(n - 2)\) 6. **Factor out the common binomial factor:** - Because both terms contain \((n - 2)\), factor this out: - \((15n + 3)(n - 2)\) 7. **Simplify the expression:** - Notice that \(15n + 3\) has a common factor of 3: - \(3(5n + 1)(n - 2)\) So, the factored form of the quadratic expression \(15n² - 27n - 6\) is: \[3(5n + 1)(n - 2)\]