Factoring Trinomials of the Form x2 + bx + c Section 6.2 Factoring Trinomials of the Form x2 + bx + c

6.2 Lecture Guide: Factoring trinomials of the form x2 + bx + c Objective 1: Factor trinomials of the form by trial-and-error method or by inspection. A trinomial in the form is called a quadratic trinomial. Your goal for this section should be to be able to factor quadratic trinomials as quickly and efficiently as possible. We will restrict our focus to the case where the leading coefficient a = 1 for this section.

To begin understanding the procedure we use to factor these trinomials, examine the following three products. Product is 1 Product is 12 Factors F O I L Product = +12 Sum is the linear term

Thus, a factorization of must have a pair of factors of c whose sum is b.

Factorable Trinomials Algebraically Verbally Example Where And . A trinomial is factorable into a pair of binomial factors with integer coefficients if and only if there are two integers whose ____________ is c and whose ____________ is b. Otherwise, the trinomial is prime over the integers. and . is prime because there are no factors of ______ whose sum is ______.

Factoring by Trial-and-Error or by Inspection To factor by trial and error list the possible factors of c and examine them until you find the right combination that produces the correct linear term. To factor by inspection, try to mentally determine two factors of c whose sum is b. Initially, it is a good idea to list the factors. This will help you to notice patterns and develop a sense for factoring trinomials. With practice you will be able to factor many of the trinomials in this section by inspection.

In problems 1-8, the values below the binomial factors represent the possible factors of the constant term c in the quadratic trinomial. Use this information along with the given sign pattern to determine the factored form. 1. 2. 1 20 2 10 4 5 1 20 2 10 4 5

3. 4. 1 30 2 15 3 10 5 6 1 30 2 15 3 10 5 6

5. 6. 1 12 2 6 3 4 1 12 2 6 3 4

7. 8. 1 36 2 18 3 12 4 9 6 1 36 2 18 3 12 4 9 6 Note: What is wrong with problem 8? How can you fix it?

Verbally Algebraically Examples If the constant term is ____________, the factors of this term must have the same sign. These factors will share the same sign as the linear term. If the constant term is ____________, the factors of this term must be opposite in sign. The sign of the constant factor with the larger absolute value will be the same as that of the linear term.

Match each trinomial with the appropriate sign pattern for the factors of this trinomial. 9. 10. 11. 12. A. B. C.

Factor each trinomial by trial-and-error or by inspection. 13. 14.

Factor each trinomial by trial-and-error or by inspection. 15. 16.

Factor each trinomial by trial-and-error or by inspection. 17. 18.

Factor each trinomial by trial-and-error or by inspection. 19. 20.

Factor each trinomial by trial-and-error or by inspection. 21. 22.

Factor each trinomial by trial-and-error or by inspection. 23. 24.

Objective 2: Identify a prime trinomial of the form . A Prime Polynomial A polynomial is prime over the integers if its only factorization must involve either ____ or _____ as one of the factors.

Use the given tables to assist you in identifying which trinomials are prime and factoring the trinomials that can be factored over the integers. Note that if we list all possibilities in a table and none of them work, then the polynomial must be prime. 25. Factors of 36 Sum of Factors 1 36 2 18 3 12 4 9 6 37 20 15 13 12

Use the given tables to assist you in identifying which trinomials are prime and factoring the trinomials that can be factored over the integers. Note that if we list all possibilities in a table and none of them work, then the polynomial must be prime. 26. Factors of 36 Sum of Factors 1 36 2 18 3 12 4 9 6 37 20 15 13 12

Use the given tables to assist you in identifying which trinomials are prime and factoring the trinomials that can be factored over the integers. Note that if we list all possibilities in a table and none of them work, then the polynomial must be prime. 27. Factors of 24 Sum of Factors −1 −24 −2 −12 −3 −8 −4 −6 −25 −14 −11 −10

Use the given tables to assist you in identifying which trinomials are prime and factoring the trinomials that can be factored over the integers. Note that if we list all possibilities in a table and none of them work, then the polynomial must be prime. 28. Factors of 24 Sum of Factors −1 −24 −2 −12 −3 −8 −4 −6 −25 −14 −11 −10

Use the given tables to assist you in identifying which trinomials are prime and factoring the trinomials that can be factored over the integers. Note that if we list all possibilities in a table and none of them work, then the polynomial must be prime. 29. Factors of 56 Sum of Factors −1 56 −2 28 −4 14 −7 8 55 26 10 1

Use the given tables to assist you in identifying which trinomials are prime and factoring the trinomials that can be factored over the integers. Note that if we list all possibilities in a table and none of them work, then the polynomial must be prime. 30. Factors of 56 Sum of Factors −1 56 −2 28 −4 14 −7 8 55 26 10 1

Objective 3: Factor trinomials of the form by the trial-and-error or by inspection. Factor each trinomial by trial-and-error or by inspection. 31.

Factor each trinomial by trial-and-error or by inspection. 32.

Factor each trinomial by trial-and-error or by inspection. 33.

Factor each trinomial by trial-and-error or by inspection. 34.

Factor each trinomial by trial-and-error or by inspection. 35.

Factor each trinomial by trial-and-error or by inspection. 36.

Each of the following trinomials has a greatest common factor Each of the following trinomials has a greatest common factor. First factor out the GCF and then complete the factorization 37.

Each of the following trinomials has a greatest common factor Each of the following trinomials has a greatest common factor. First factor out the GCF and then complete the factorization 38.

Each of the following trinomials has a greatest common factor Each of the following trinomials has a greatest common factor. First factor out the GCF and then complete the factorization 39.

Each of the following trinomials has a greatest common factor Each of the following trinomials has a greatest common factor. First factor out the GCF and then complete the factorization 40.

Each of the following trinomials has a greatest common factor Each of the following trinomials has a greatest common factor. First factor out the GCF and then complete the factorization 41.

Each of the following trinomials has a greatest common factor Each of the following trinomials has a greatest common factor. First factor out the GCF and then complete the factorization 42.

Each of the following trinomials has a greatest common factor. First factor out the GCF and then complete the factorization. 43.

Each of the following trinomials has a greatest common factor. First factor out the GCF and then complete the factorization. 44.

Each of the following trinomials has a greatest common factor. First factor out the GCF and then complete the factorization. 45.

Each of the following trinomials has a greatest common factor. First factor out the GCF and then complete the factorization. 46.

x-intercepts of the graph of Complete the table for each polynomial. Polynomial Factored Form Zeros of x-intercepts of the graph of 47. 48.