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Factoring Trinomials, Part 2 Section 6.3&4 Factoring Trinomials, Part 2

Review from yesterday’s homework: 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(x2 + 10x + 24) = 3x4(x + 4)(x + 6).

There is no GCF that can be factored out. In all of yesterday’s homework problems, the leading coefficient of the trinomial was 1, or became 1 after the GCF was factored out, as in the review problem we just did. But what if the leading coefficient is NOT 1, and is still larger than 1 after any GCF is factored out? In some cases, the resulting trinomial is prime, but in many other cases, it can be factored into two binomials. Example: Factor 3x2 + 14x + 8. There is no GCF that can be factored out. However, this polynomial is NOT prime. The factors of this trinomial are (3x + 2)(x+4). But how do we figure this out?

3*8 = 24 Back to the problem: Factor 3x2 + 14x + 8. We’re going to show you a method for these kinds of problems that may be easier for you than the “guess and check” method shown in some of the textbook/online examples. This method is sometimes called the “British Method”, and it starts with splitting the middle term of the trinomial into two parts, so that we can use factoring by grouping on the resulting four-term polynomial. First: multiply the first and last coefficients of the trinomial together: 3*8 = 24 Now look for two factors of 24 that add up to 14 (the middle number): 24 = 3*8, but 3 + 8 = 11 24 = 6*4, but 6 + 4 = 10 24 = 2*12, and 2 + 12 = 14, which is what we’re looking for. Now split the middle term, 14x, into two pieces, 2x + 12x This now gives 3x2 + 2x + 12x + 8 Now we have a FOUR-TERM polynomial. What does this suggest??

You got it: Factoring by grouping! 3x2 + 2x + 12x + 8 Factoring the first pair gives: 3x2 + 2x = x(3x + 2) Factoring the 2nd pair gives: 12x + 8 = 4(3x + 2) Putting it all back together now gives: x(3x+2) + 4(3x + 2) = (3x + 2)(x+4) Done! Don’t forget to always check your answer by multiplying it back out. (You should get in the habit of doing this on your homework so you don’t forget to do it on quizzes.)

Factor the polynomial 25x2 + 20x + 4. Example Factor the polynomial 25x2 + 20x + 4. First, always check for a GCF. There isn’t one, so now we will multiply the first and last coefficients together: 25*4 = 100 Now find two factors of 100 that add up to 20: 2* 50? Nope (2 + 50 = 52) 5*20? Nope (5 + 20 = 25) 4*25? Nope (4 + 25 = 29) 10*10? Bingo! (10 + 10 = 20) Use this to split the middle term (20x) into two pieces: 25x2 + 10x + 10x + 4. Now factor by grouping. Answer: (5x + 2)(5x + 2) (check it!) What’s a better way to write this? (5x + 2)2

3*8 = 24 But consider this example: Factor 3x2 + 15x + 8. In the examples we’ve done so far, it looks like the “factoring by grouping” method always works for a trinomial with a leading coefficient other than one that can’t be factored away. But consider this example: Factor 3x2 + 15x + 8. First: multiply the first and last coefficients of the trinomial together: 3*8 = 24 Now look for two factors of 24 that add up to 15. product: sum: 24 = 1*24 1+ 24 = 25 24 = 2*12 2 + 12 = 14 24 = 3*8 3 + 8 = 11 24 = 6*4 6 + 4 = 10 There are no other ways to factor 24, and none of our factor pairs add up to 15, so this polynomial is prime.

Example Factor the polynomial 21x2 – 41x + 10. 1. Multiply 21*10 = 210 2. Find 2 factors of 210 that add up to -41 [Note: 210 = 2*3*5*7] -10*-21? What do these factors add up to? -2*-105? -5*-42? -15*-14? -3*-70? -6*-35? BINGO! Now factor by grouping: 21x2 - 6x – 35x+10 3x(7x – 2) -5(7x – 2) = (7x +2)(3x – 5) NOW CHECK IT!

Example Factor the polynomial 3x2 + 20x – 4. Step 1: 3*-4 = -12 Step 2: possible factors of -12: 3*-4 sum = -1 -3*4 sum = 1 -1*12 sum = 11 1*-12 sum = -11 -2*6 sum = 4 2* -6 sum = -4 There are no other possible ways to factor -12, and none of the combinations add up to 20, so this is a PRIME polynomial (it can’t be factored).

Example Factor the polynomial 6x2y2 – 10xy2 – 4y2. Remember to ALWAYS check for a GCF first. The GCF of the three terms of this polynomial is 2y2, so we factor that out first: 6x2y2 – 10xy2 – 4y2 = 2y2(3x2 – 5x – 2) Now factor (3x2 -5x -2) by the “factoring by grouping” method shown in the previous examples. Final answer: 2y2(3x + 1)(x - 2) (remember to check!)

Weekly Quiz 8 tomorrow on HW 31-33 AND + WeeklyQuiz7 questions This quiz is worth 10 points, will have 10 questions and a 25-minute time limit, and will be given in the last half of class, after the lecture. The practice quiz has 15 questions and a 40-minute time limit. You can take the practice quiz as many times as you want, and only your best score will count towards your overall course grade. The practice quiz is a required 4-point assignment, and is due at the start of class tomorrow, along with HW 34.

Mondays through Thursdays 8:00 a.m. to 7:30 p.m. REMINDERS: The assignment on today’s material (HW 34) is due at the start of the next class session. Practice Weekly Quiz 8 is also due at the start of class tomorrow. Lab hours in 203: Mondays through Thursdays 8:00 a.m. to 7:30 p.m. Please remember to sign in on the Math 110 clipboard by the front door of the lab

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 quiz/test.