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Chapter 7: Problem 4

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 \(n ?\)

Short Answer

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

Step by step solution

01

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.
02

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 \(x^2 + nx + mx + mn\).
03

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) \(mn > 0\)
04

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.
05

Conclusion

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

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