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Chapter 0: Problem 142

Find all integers \(b\) so that the trinomial can be factored. $$x^{2}+b x+15$$

Short Answer

Expert verified
The integer values of \(b\) that make the trinomial factorable are -11, -9, -7, -5, 5, 7, 9, 11.

Step by step solution

01

Express the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Here \(b\) is unknown and \(D\) should be a perfect square for the quadratic equation to be factorable. Write the expression \(D = b^2 - 4*1*15 = b^2 - 60\).
02

Find Perfect Squares

For the equation to be factorable, the discriminant should equal a perfect square. So, find the perfect squares which are less than or equal to 60. These are: 0, 1, 4, 9, 16, 25, 36, 49.
03

Solve for \(b\)

Set each of the perfect square values found in Step 2 equal to the discriminant (which should also be a perfect square) and solve for \(b\). This will produce two values of \(b\) for each equation 鈥� one positive and one negative. Remember, since \(b\) is an integer, discard all non-integer solutions.
04

List All Possible Integer Values of \(b\)

After solving, the possible integer solutions for \(b\) are: -11, -9, -7, -5, 5, 7, 9, 11.

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