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General Trinomials
 

x2 + (a + b)x + ab = (x + a)(x + b)


The expression x2 +(a + b)x + ab tells us that we need two numbers whose product is the last term and whose sum is the coefficient of the middle term. Consider the trinomial x2 + 5x + 6 . Now, two factors of 6 are 1 and 6, but 1 + 6 ≠ 5. However, 2 and 3 are also factors of 6, and 2 + 3 = 5. Hence, x2 + 5x + 6 = (x + 2)(x + 3).

 

Example:

Column A
x

Column B
7

where x2 - 7x - 18 = 0
 

Now, both 2 and ––9 are factors of 18, and 2 + (––9) = ––7. Hence, x2 - 7x - 18 =(x + 2)(x - 9)= 0 . Setting each factor equal to zero yields x + 2 = 0 and x –– 9 = 0. Solving these equations yields x = ––2 and 9. If x = ––2, then Column B is larger. However, if x = 9, then Column A is larger. This is a double case, and the answer is (D).

 

Complete Factoring
 

When factoring an expression, first check for a common factor, then check for a difference of squares, then for a perfect square trinomial, and then for a general trinomial.

 

Example:
Factor the expression 2x3 - 2x 2 -12x completely.

Solution:
First check for a common factor: 2x is common to each term. Factoring 2x out of each term yields 2x(x2 - x - 6). Next, there is no difference of squares, and x2 - x - 6 is not a perfect square trinomial since x does not equal twice the product of the square roots of x2 and 6. Now, ––3 and 2 are factors of ––6 whose sum is ––1. Hence, 2x(x2 - x - 6) factors into 2x(x –– 3)(x + 2).





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