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Chapter 0: Problem 143

Find all integers \(b\) so that the trinomial can be factored. $$x^{2}+4 x+b$$

Short Answer

Expert verified
The possible integer values for \(b\) that factorize the trinomial are 4, 3, 2, 1, 0.

Step by step solution

01

Calculate the Discriminant

First, calculate the square of the middle coefficient and subtract it from four times the last coefficient. For this particular case, it will give us \(16-4b\). However, we need \(-B \pm \sqrt{D}\) to be integers, so the discriminant D, i.e., \(16-4b\), must be a perfect square. Let's represent it with \(m^2\).
02

Solve for \(b\)

Next, set the expression equal to \(m^2\) to solve for the potential values of \(b\). Meaning \(16-4b=m^2\). If you rearrange this equation to solve for \(b\), you get \(b=4-m^2\). Since we are asked for integer values of \(b\), \(m^2\) should be an integer that is less than or equal to 4.
03

Identify the integer solutions

Finally, determine the integers less than or equal to 4. These are \(m^2 = -2, -1, 0, 1, 2\). These values for \(m^2\) produce corresponding \(b\) values to be \(b=4, 3, 2, 1, 0\) respectively. Thus, all integer solutions for \(b\) are 4, 3, 2, 1, 0.

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