Factoring Trinomials, Part 1 Section 6.2 Factoring Trinomials, Part 1

Review problem from Section 6 Review problem from Section 6.1: (Factoring out the GCF and factoring 4-term polynomials by grouping) Factor completely: Answer: How would you check this? (Because you WOULD do that, wouldn’t you???)

Section 6.2 Factoring Trinomials, Part 1 Recall by using the FOIL method that (x + 2)(x + 4) = x2 + 4x + 2x + 8 = x2 + 6x + 8 So to factor x2 + 6x + 8 into (x + __ ) (x + __ ), note that 6 is the sum of the two numbers 4 and 2, and 8 is the product of the two numbers. So we’ll be looking for 2 numbers whose product is 8 and whose sum is 6. Note: there are fewer possible pairs of numbers for the product than for the sum, so that’s why we start there first.

Factor the polynomial x2 + 13x + 30. Example Factor the polynomial x2 + 13x + 30. Since our two numbers must have a product of 30 and a sum of 13, the two numbers must both be positive. Positive factors of 30 Sum of Factors 1, 30 1+30=31 2, 15 2+15=17 3, 10 3+10=13 Note, there are other factors (like 6*5), but once we find a pair that works, we do not have to continue searching. So x2 + 13x + 30 = (x + 3)(x + 10). Now check answer by multiplying the two factors to see if you get back to the original trinomial.

Factor the polynomial x2 – 11x + 24. Example Factor the polynomial x2 – 11x + 24. Since our two numbers must have a product of 24 and a sum of -11, the two numbers must both be negative. Negative factors of 24 Sum of Factors -1, -24 -25 -2, -12 -14 -3, -8 -11 So x2 – 11x + 24 = (x – 3)(x – 8). Now check it!

Factor the polynomial x2 – 2x – 35. Example Factor the polynomial x2 – 2x – 35. Since our two numbers must have a product of -35 and a sum of -2, the two numbers will have to have different signs. Factors of -35 Sum of Factors -1, 35 34 1, -35 -34 -5, 7 2 5, -7 -2 So x2 – 2x – 35 = (x + 5)(x – 7). Check it!

Factor the polynomial x2 – 6x + 10. Example Factor the polynomial x2 – 6x + 10. Since our two numbers must have a product of 10 and a sum of -6, the two numbers will have to both be negative. Negative factors of 10 Sum of Factors -1, -10 -11 -2, -5 -7 Now we have a problem, because we have exhausted all possible choices for the factors, but have not found a pair whose sum is -6. So x2 – 6x +10 is not factorable and we call it a prime polynomial.

Factor the polynomial x2 – 11xy + 30y2. Example Factor the polynomial x2 – 11xy + 30y2. We look for two terms whose product is 30x2y2 and whose sum is –11xy. The two terms will have to both be negative. Note: each term will contain the variable y, for the sum to be –11xy. Negative factors of 30x2y2 Sum of Factors -xy, -30xy -31xy -2xy, -15xy -17xy -3xy, -10xy -13xy -5xy, -6xy -11xy So x2 – 11xy + 30y2 = (x – 5y)(x – 6y).

Factor the polynomial 3x6 + 30x5 + 72x4 Example Factor the polynomial 3x6 + 30x5 + 72x4 First we factor out the GCF. (Always check for this first!) 3x6 + 30x5 + 72x4 = 3x4(x2 + 10x + 24) Then we factor the trinomial. Positive factors of 24 Sum of Factors 1, 24 25 2, 12 14 3, 8 11 4, 6 10 So 3x6 + 30x5 + 72x4 = 3x4(x + 4)(x + 6).

REMINDER: On a test or quiz, if you have time left after finishing all the problems, you should always check your factoring results by multiplying the factored polynomial to verify that it is equal to the original polynomial. Many times you can detect computational errors or errors in the signs of your numbers (i.e those pesky “dumb mistakes”…) by checking your results. Practice doing this on at least a few homework problems before you hit “Check Answer”, just to make sure you really do know how to check your answers when it comes time for the quiz.

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and begin working on the homework assignment. You may now OPEN your LAPTOPS and begin working on the homework assignment. We expect all students to stay in the classroom to work on your homework till the end of the 55-minute class period. If you have already finished the homework assignment for today’s section, you should work ahead on the next one or work on the next practice test.