SOLUTION: Find the number of possible values of p so that 4x2+ px-9 can be factored as the product of two integral first degree polynomials.

Question 1209266: Find the number of possible values of p so that 4x2+ px-9 can be factored as the product of two integral first degree polynomials.

Answer by asinus(45) (Show Source): You can put this solution on YOUR website!
**1. Understand the Problem**
* We're given a quadratic equation: 4x² + px - 9
* We need to find the possible integer values of 'p' that allow this equation to be factored into two binomials with integer coefficients.
**2. Factor the Quadratic (General Form)**
* Assume the quadratic can be factored as: (ax + b)(cx + d)
* where a, b, c, and d are integers.
* Expanding this gives: acx² + (ad + bc)x + bd
* Comparing this to the given quadratic:
* ac = 4
* ad + bc = p
* bd = -9
**3. Find Possible Combinations**
* **Find possible values for 'a' and 'c':** Since ac = 4, the possible combinations for 'a' and 'c' are:
* (a, c) = (1, 4), (4, 1), (2, 2), (-1, -4), (-4, -1), (-2, -2)
* **Find possible values for 'b' and 'd':** Since bd = -9, the possible combinations for 'b' and 'd' are:
* (b, d) = (1, -9), (-1, 9), (3, -3), (-3, 3), (9, -1), (-9, 1)
**4. Test Combinations and Find Possible Values of 'p'**
* For each combination of (a, c) and (b, d), calculate the value of 'p' using the equation 'p = ad + bc'.
* Keep only the values of 'p' that are integers.
**5. Determine the Number of Possible Values of 'p'**
* Count the number of distinct integer values of 'p' obtained in step 4.
**Example (Illustrative)**
* Let's consider one combination: (a, c) = (1, 4) and (b, d) = (1, -9)
* p = ad + bc = (1 * -9) + (4 * 1) = -5
* Repeat this process for all possible combinations of 'a', 'c', 'b', and 'd'.
**Note:** This method involves systematically checking all possible combinations of factors, which can be time-consuming. You can use a spreadsheet or a programming approach to automate the calculations.
By following these steps, you can determine the number of possible integer values of 'p' that allow the given quadratic equation to be factored into two binomials with integer coefficients.

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