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

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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 $$c$$.
To make the quadratic expression $$x^2 + bx + c$$ factorable with integer coefficients, $$c$$ must be such that it can be expressed as the product of two integers that add up to $$b$$. The smallest integer for $$c$$ is typically negative, while the largest is positive and depends on the factor pairs of $$ac$$.

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

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