The diamond method provides another systematic way to factor second degree polynomials of the form ax^{2} + bx + c that are "hard," that is, those second degree polynomials that have a value for *a* other than 1.

This method is best illustrated using an example.

Suppose we want to factor 2x^{2} - 5x - 3.

1) First we draw an "X" shape, our diamond. We put the value of a*c in the top, and b in the bottom.

2) Next, we consider what factors of the value in the top of the diamond (-6) multiply to the value in the top of the diamond (-6) as well as add up to the value in the bottom of the diamond (-5). The factors of -6 are:

1 and -6,

-1 and 6,

2 and -3,

-2 and 3

In this case the factors that add up to -5 are 1 and -6. We place these values in the left and right parts of the diamond (does not matter which one goes where).

3) Now we will make a fraction out of the left and right values, making the existing values (the -6 and 1) the denominators, and the value of *a* (2) as the coefficient of x the numerator.

4) We reduce these fractions. The numerator value represents the variable part of the coefficient, and the denominator represents the number part of the coefficient.

5) This gives us binomial factors of (x - 3) and (2x + 1). Check that they are correct by multiplying them to see if we get our original polynomial:

(x - 3)(2x + 1) = 2x^{2} - 5x -3, which is our originial polynomial.

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