Answer
The expression \( 912m^2 - 19mn + 5n^2 \) cannot be factored into simpler binomial terms with integer coefficients. It is **irreducible** over the integers.
Solution
The expression you've provided is a quadratic in terms of variables \( m \) and \( n \):
\[
912m^2 - 19mn + 5n^2
\]
### **Objective: Factor the Expression**
To factor this quadratic expression, we look for two binomials of the form:
\[
(am + bn)(cm + dn)
\]
When expanded, this yields:
\[
ac\,m^2 + (ad + bc)\,mn + bd\,n^2
\]
By comparing coefficients, we establish the following system of equations:
1. \( ac = 912 \) *(Coefficient of \( m^2 \))*
2. \( ad + bc = -19 \) *(Coefficient of \( mn \))*
3. \( bd = 5 \) *(Coefficient of \( n^2 \))*
#### **Step 1: Find Possible Pairs for \( a \) and \( c \)**
First, factorize 912:
\[
912 = 2^4 \times 3 \times 19
\]
Possible integer pairs \((a, c)\) such that \( ac = 912 \):
- \( (1, 912) \)
- \( (2, 456) \)
- \( (3, 304) \)
- \( (4, 228) \)
- \( (6, 152) \)
- \( (8, 114) \)
- \( (12, 76) \)
- \( (16, 57) \)
- \( (19, 48) \)
- \( (24, 38) \)
- ...and their negative counterparts.
#### **Step 2: Find Possible Pairs for \( b \) and \( d \)**
Since \( bd = 5 \) and 5 is a prime number, the possible integer pairs are:
- \( (1, 5) \)
- \( (-1, -5) \)
#### **Step 3: Check for Compatibility**
For each combination of \((a, c)\) and \((b, d)\), check if \( ad + bc = -19 \).
**Example Attempt:**
Let's try \( b = 1 \) and \( d = 5 \):
1. \( ad + bc = a \times 5 + c \times 1 = 5a + c = -19 \)
2. From \( ac = 912 \), suppose \( a = 19 \), then \( c = 48 \).
3. Check \( 5a + c = 5 \times 19 + 48 = 95 + 48 = 143 \neq -19 \)
This doesn't satisfy the second equation. Repeating this process for all possible pairs reveals that **no integer solutions satisfy all three equations simultaneously**.
### **Conclusion**
The quadratic expression:
\[
912m^2 - 19mn + 5n^2
\]
**is not factorable** into binomials with integer coefficients. This means the expression is **irreducible** over the set of integers.
### **Alternative Approach: Using the Quadratic Formula**
If you consider the expression as a quadratic in \( m \), you can explore its roots using the quadratic formula:
\[
m = \frac{19n \pm \sqrt{(19n)^2 - 4 \times 912 \times 5n^2}}{2 \times 912} = \frac{19n \pm \sqrt{-17879n^2}}{1824}
\]
The discriminant here is negative (\( -17879n^2 \)), indicating that the quadratic does not have real roots. This further confirms that the expression cannot be factored into real linear factors.
### **Final Answer**
The expression \( 912m^2 - 19mn + 5n^2 \) cannot be factored into simpler binomial terms with integer coefficients. It is **irreducible** over the integers.
Reviewed and approved by the UpStudy tutoring team
Explain
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