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FACTORING TRINOMIALS

Factoring is the opposite of generation. Since the product of two binomials is normally a trinomial, we can expect factorable trinomials ( that have terms with no park factor ) to have two binomial factors.Thus, factoring trinomials requires using FOIL backwards.
Factor each trinomial .

To factor this polynomial, we must find integers a, bel, coulomb, and d such that

4y^2 – 11y + 6 = ( ay + b ) ( cy + vitamin d ) .
By using FOIL, we see that ac = 4 and bd = 6. The positive factors of 4 are 4 and 1 or 2 and 2. Since the middle terminus is negative, we consider lone negative factors of 6. The possibilities are – 2 and – 3 or – 1 and – 6. immediately we try respective arrangements of these factors until we find one that gives the chastise coefficient of y .

The last test gives the right factorization.
divisor

again, we try respective possibilities. The positive factors of 6 could be 2 and 3 or 1 and 6. As factors of – 5 we have merely -1 and 5 or – 5 and 1. Try unlike combinations of these factors until the correct one is found .

finally, 6p^2 – 7p – 5 factors as ( 3p – 5 ) ( 2p + 1 )
In the above examples, we chose positive factors of the positive first condition. Of course, we could have used two negative factors, but the work is easier if positivist factors are used .
Each of the extra patterns of generation given earlier can be used in turn back to get a pattern for factoring. Perfect square trinomials can be factored as follows .

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PERFECT SQUARE TRINOMIALS
 


These formulas should be memorized .
Factor each polynomial .

Since 16p^2 = ( 4p ) ^2 and 25q^2 = ( 5q ) ^2, use the irregular practice shown above with 4p replacing adam and 5q replacing yttrium to get

Make surely that the center condition of the trinomial being factored, -40pq here, is doubly the merchandise of the two terms in the binomial 4p – 5q .

FACTORING BINOMIALS

The practice for the merchandise of the summarize and deviation of two terms gives the follow factorization .

Factor each of the following polynomials .
Example 1

first, recognize that 4m^2 – 9 is the dispute of two squares, since 4m^2 = ( 2m ) ^2 and 9 = 3^2. Use the blueprint for the difference of two squares with 2m replacing ten and 3 replacing y. Doing this gives :

Example 2

Use the deviation of two squares pattern doubly, as follows :

Example 3

Group the first three terms to get a perfect square trinomial. then use the dispute of squares pattern .

Two other extra results of factoring are listed below. Each can be verified by multiplying on the right side of the equality .

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Factor each polynomial .
Example 1

Notice that 27 = 3^3, so the expression is a sum of two cubes. Use the irregular form given above .

Example 2

Since 64n^3 = ( 4n ) ^3, the given polynomial is a deviation of two cubes. To factor, use the beginning pattern in the box above, replacing adam with thousand and yttrium with 4n .

Example 3

write 8q^6 as ( 2q^2 ) ^3 and 125p9 as ( 5p^3 ) ^3, so that the given polynomial is a sum of two cubes .

METHOD OF SUBSTITUTION

sometimes a polynomial can be factored by substituting one saying for another. The adjacent exercise shows this method acting of substitution .
Factor each polynomial .

( Some students prefer to factor this type of trinomial immediately using trial and error with FOIL. )

now replace molarity with 2a – 1 in the factor shape and simplify .