Factoring Trinomials of the Form ax2 + bxy + cy2 Section 6.3 Factoring Trinomials of the Form ax2 + bxy + cy2

7.2 Lecture Guide: Factoring Trinomials of the Form ax2 + bxy + cy2 Objective 1: Factor trinomials of the form Objective 2: Factor trinomials of the form

Factoring a polynomial can be considered a reversal of the process of multiplying the factors of the polynomial. In Section 6.2, we focused on factoring trinomials where the leading coefficient was 1. Factoring trinomials where the leading coefficient is not 1 can be more complicated. We will start by multiplying several pairs of factors that form a trinomial with a leading coefficient of 6.

1. Multiply the factors in this table by writing out both middle terms and then simplify the result. The first row has been completed. Factors FIRST MIDDLE LAST Products

1. Multiply the factors in this table by writing out both middle terms and then simplify the result. The first row has been completed. Factors FIRST MIDDLE LAST Products

2. Answer each question about the table above. (a) What is the product of the coefficients of each pair of middle terms? (b) What do you notice about the first and last term of each product? (c) What is the product of the coefficients of the first and last terms? (d) What is the correct factorization of ? (e) The procedure for factoring trinomials of the form by tables and grouping involves finding two factors of ac whose sum is b. When expanded, the correct factorization of has two middle terms whose coefficients have a product of ____________ and a sum of ____________.

Step 1: Be sure you have factored out the GCF if it is not 1. Factoring by Tables and Grouping Example Procedure Step 1: Be sure you have factored out the GCF if it is not 1. (If a<0, factor out -1.) Factor Factors of –120 –1 –2 –3 –4 –5 –6 –8 –10 Sum of Factors Step 2: Find two factors of ac whose sum is b. Note: If ac is positive then both factors must have the __________ __________. If ac is negative, then the two factors will have the ___________ sign with the “larger” factor having the same sign as b.

Factoring by Tables and Grouping Example Procedure Step 3: Rewrite the linear term of ax2 + bx + c so that b is the sum of the pair of factors from Step 2. Step 4: Factor the polynomial from Step 3 by grouping the terms and factoring the GCF out of each pair of terms. Example Factors of –120 –8 Sum of Factors

Factor each polynomial using the method of tables and grouping. Factors of ___ Sum of Factors 3. Multiply the factors to check your work.

Factor each polynomial using the method of tables and grouping. Factors of ___ Sum of Factors 4. Multiply the factors to check your work.

Factor each polynomial using the method of tables and grouping. Factors of ___ Sum of Factors 5. Multiply the factors to check your work.

Fill in the missing information to complete the factorization of each trinomial by inspection. 6. 7. 8. 9.

Fill in the missing information to complete the factorization of each trinomial by inspection. 10. 11. 12. 13.

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection. 14.

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection. 15.

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection. 16.

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection. 17.

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection. 18.

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection. 19.

Factor each trinomial by inspection or by tables and grouping Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result. 20.

Factor each trinomial by inspection or by tables and grouping Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result. 21.

Factor each trinomial by inspection or by tables and grouping Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result. 22.

Factor each trinomial by inspection or by tables and grouping Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result. 23.

Factor each trinomial by inspection or by tables and grouping Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result. 24.

Factor each trinomial by inspection or by tables and grouping Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result. 25.

Remember to first factor out the GCF. 26.

Remember to first factor out the GCF. 27.

Remember to first factor out the GCF. 28.

Remember to first factor out the GCF. 29.

Remember to first factor out the GCF. 30.

Remember to first factor out the GCF. 31.

Remember to first factor out the GCF. 32.

Remember to first factor out the GCF. 33.