1) Factoring Quadratics With Undefined C Fill in the blanks by finding the largest and smallest integers that will make the quadratic expression factorable. Explain your reasoning and show your working

To determine the largest and smallest integers that will make a quadratic expression factorable, we need to consider a general quadratic expression of the form: \[ ax^2 + bx + c \] In this case, we are given that \(c\) is undefined, meaning we need to find conditions on \(c\) such that the quadratic can be factored into two binomials. ### Step 1: Understanding Factorability A quadratic expression \(ax^2 + bx + c\) is factorable over the integers if there exist two integers \(m\) and \(n\) such that: \[ m + n = b \quad \text{and} \quad m \cdot n = ac \] ### Step 2: Setting Up the Problem Since \(c\) is undefined, we can express \(c\) in terms of \(m\) and \(n\): \[ c = \frac{m \cdot n}{a} \] ### Step 3: Finding Integer Values To find the largest and smallest integers for \(c\), we need to consider the following: 1. **Factor pairs of \(ac\)**: The product \(m \cdot n\) must equal \(ac\). Therefore, we need to find pairs of integers that multiply to give \(ac\). 2. **Sum of the pairs**: The sum of these pairs must equal \(b\). ### Step 4: Example Calculation Let's assume \(a = 1\) for simplicity, so we have: \[ x^2 + bx + c \] We need to find integers \(m\) and \(n\) such that: \[ m + n = b \quad \text{and} \quad m \cdot n = c \] ### Step 5: Finding Limits for \(c\) To find the largest and smallest integers for \(c\): - **Smallest \(c\)**: This occurs when \(m\) and \(n\) are both negative or both positive and as close to zero as possible. For example, if \(b = 0\), then \(m = 1\) and \(n = -1\) gives \(c = -1\). - **Largest \(c\)**: This occurs when \(m\) and \(n\) are both large positive integers. For example, if \(b = 10\), then \(m = 5\) and \(n = 5\) gives \(c = 25\). ### Conclusion To summarize, the largest and smallest integers for \(c\) depend on the values of \(b\) and the factor pairs of \(ac\). - The smallest integer \(c\) can be is typically negative, depending on the values of \(m\) and \(n\). - The largest integer \(c\) can be is positive and can be determined by the maximum product of the factor pairs. If you have specific values for \(a\) and \(b\), I can provide a more precise calculation for the largest and smallest integers for \(c\).