SOLUTION: Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions: 2mn - 18m + 5n

SOLUTION: Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions: 2mn - 18m + 5n - mn + 20m + 4n + ___

Question 1209739: Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions:
2mn - 18m + 5n - mn + 20m + 4n + ___
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's how to find the constant:
1. **Combine Like Terms:**
First, combine the like terms in the expression:
2mn - 18m + 5n - mn + 20m + 4n + C = mn + 2m + 9n + C
2. **Factoring Pattern:**
We want this expression to be factorable into the form (m + a)(n + b) for some constants *a* and *b*. Expanding this factored form gives us:
mn + bm + an + ab
3. **Match Coefficients:**
Compare the coefficients of the terms in our simplified expression (mn + 2m + 9n + C) with the coefficients in the expanded factored form (mn + bm + an + ab):
* The coefficient of mn is 1 in both.
* The coefficient of m is 2, so b = 2.
* The coefficient of n is 9, so a = 9.
* The constant term is C, and it must equal ab.
4. **Solve for C:**
Since a = 9 and b = 2, we have:
C = ab = (9)(2) = 18
Therefore, the constant is 18. The factored expression is (m+9)(n+2).

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