Problem 4 - Beginning and Intermediate Algebra

Question

If \(x^{2}+b x+c\) factors to \((x+m)(x+n)\) and if \(b\) and \(c\) are positive, what do you know about the signs of \(m\) and \(n ?\)

Step-by-Step Solution

Answer

Both \(m\) and \(n\) are positive, as the product and sum of their coefficients are positive.

Step 1: Write out given information

We are given that the quadratic equation \(x^2 + bx + c\) factors to \((x+m)(x+n)\) and that \(b\) and \(c\) are positive.

Step 2: Expand the factored quadratic equation and equate coefficients

Expand the factored equation \((x+m)(x+n)\) to its standard form by applying the distributive property: \((x+m)(x+n) = x^2 + nx + mx + mn\). Now, we can equate the corresponding coefficients of the given equation \(x^2 + bx + c\) and the expanded equation \(x^2 + nx + mx + mn\).

Step 3: Analyze the signs of the coefficients of the expanded equation

From Step 2, we have the following two equations: 1) \(n + m = b\) (since \(nx + mx = bx\)) 2) \(mn = c\) (since \(mn\) is the constant term) Since \(b\) and \(c\) are positive, we know the following: 1) \(n + m > 0\) 2) \(mn > 0\)

Step 4: Determine the signs of \(m\) and \(n\) based on the properties of multiplication

From equation (2), \(mn > 0\). This implies that either both \(m\) and \(n\) are positive, or both \(m\) and \(n\) are negative. Since \(n + m > 0\), if both \(m\) and \(n\) were negative, their sum would be negative as well. But this contradicts the fact that \(n + m > 0\). Therefore, both \(m\) and \(n\) must be positive.

Step 5: Conclusion

Based on the given information and the properties of multiplication, we conclude that both \(m\) and \(n\) are positive.