8-6 Factoring a x ² + bx + c

Objective

To factor trinomials of the form a x 2 + b x + c eh x squared , plus b x plus c

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Essential Understanding You can write some trinomials of the form a x 2 + b x + c eh , x squared , plus b x plus c  as the product of two binomials.

Consider the trinomial 6 x 2 + 23 x + 7 . 6 , x squared , plus 23 x plus 7 .  To factor it, think of 23 x 23 x  as 2 x + 21 x . 2 x plus 21 x .

6 x 2 + 23 x + 7 = 6 x 2 + 2 x + 21 x + 7 Rewrite  23 x  as  2 x + 21 x .   = 2 x ( 3 x + 1 ) + 7 ( 3 x + 1 ) Factor out the GCF of each pair of terms.   = ( 2 x + 7 ) ( 3 x + 1 ) Distributive Property table with 3 rows and 4 columns , row1 column 1 , 6 , x squared , plus 23 x plus 7 , column 2 equals , column 3 6 , x squared , plus 2 x plus 21 x plus 7 , column 4 cap rewrite . 23 x , as , 2 x plus 21 x . , row2 column 1 , , column 2 equals , column 3 2 x open 3 x plus 1 close plus 7 open 3 x plus 1 close , column 4 cap factor out the cap gcap ccap f of each pair of terms. , row3 column 1 , , column 2 equals , column 3 open 2 x plus 7 close open 3 x plus 1 close , column 4 cap distributive cap property , end table

How do you know to rewrite 23x as 2 x + 21 x ? 2 x plus 21 x question mark  Notice that multiplying 2 and 21 gives 42, which is the product of the x 2 x squared -coefficient 6 and the constant term 7. This example suggests that, to factor a trinomial of the form a x 2 + b x + c , eh x squared , plus b x plus c comma  you should look for factors of the product ac that have a sum of b.

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