Section P.6 Factoring Trinomials 61 Factoring Trinomials of the Form Try covering the factored forms in the left-hand column below. Can you deter- mine the factored forms from the trinomial forms? Factored Form F O I L Trinomial Form Your goal here is to factor trinomials of the form To begin, consider the factorization By multiplying the right-hand side, you obtain the following result. Sum of Product of terms terms So, to factor a trinomial into a product of two binomials, you must find two factors of c with a sum of b . Example 1 Factoring Trinomials Factor the trinomials (a) and (b) Solution (a) You need to find two factors whose product is and whose sum is The product of and 2 is The sum of and 2 is (b) You need to find two factors whose product is 6 and whose sum is The product of and is 6 The sum of and is Now try Exercise 7. Note that when the constant term of the trinomial is positive, its factors must have like signs; otherwise, its factors have unlike signs. 5. 2 3 x 2 5 x 6 x 3 x 2 2 3 5. 2. 4 x 2 2 x 8 x 4 x 2 8. 4 2. 8 x 2 5 x 6. x 2 2 x 8 x 2 bx c b x c x 2 x 2 m n x mn x m x n x 2 nx mx mn x 2 bx c x m x n . x 2 bx c . 3 x 5 x 1 3 x 2 3 x 5 x 5 3 x 2 8 x 5 x 3 x 2 x 2 2 x 3 x 6 x 2 5 x 6 x 1 x 4 x 2 4 x x 4 x 2 3 x 4 x 2 bx c Factoring Trinomials P.6 • Factor trinomials of the form • Factor trinomials of the form • Factor trinomials by grouping • Factor perfect square trinomials • Select the best factoring technique using the guidelines for factoring polynomials The techniques for factoring trinomials will help you in solving quadratic equations. ax 2 bx c x 2 bx c What you should learn: Why you should learn it: Study Tip Use a list to help you find the two numbers with the required product and sum. For Example 1(a): Factors of Sum 1, 8 7 2, 4 2 Because is the required sum, the correct factorization is x 2 2 x 8 x 4 x 2 . 2 2, 2 4 1, 7 8 8
62 Chapter P Prerequisites When factoring a trinomial of the form if you have trouble finding two factors of c with a sum of b , it may be helpful to list all of the distinct pairs of factors and then choose the appropriate pair from the list. For instance, consider the trinomial For this trinomial, and So, you need to find two factors of with a sum of as shown at the left. With experience, you will be able to narrow this list down mentally to only two or three possibilities whose sums can then be tested to determine the correct factorization, which is Example 2 Factoring a Trinomial Factor the trinomial Solution To factor this trinomial, you need to find two factors whose product is and whose sum is The product of 2 and is The sum of 2 and is Now try Exercise 11. Applications of algebra sometimes involve trinomials that have a common monomial factor. To factor such trinomials completely, first factor out the common monomial factor. Then try to factor the resulting trinomial by the methods given in this section.
- Fall '19
- Algebra, Integer factorization, bx c, Factor trinomials of the form