Factoring Trinomials Factor the trinomial. 3 x^2-16 x+5 |...

Name: Problem 19, Chapter 0: Algebra and Trigonometry
Uploaded: 2019-12-27T01:32:33-08:00
Duration: 14 min 39 s
Channel: Michael Lubich
Description: Factoring Trinomials Factor the trinomial. 3 x^2-16 x+5

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically written in the form ax² + bx + c. Understanding their structure is essential for various algebraic methods, including factoring and solving quadratic equations.

Factoring Quadratics

Factoring quadratics involves rewriting the quadratic expression as a product of two binomials. This technique is crucial in simplifying expressions, solving equations, and understanding the roots or zeroes of the quadratic function.

AC Method

The AC method is a systematic technique used in factoring quadratic trinomials, especially when the coefficient of x² is not 1. It involves multiplying the coefficient of x² (a) and the constant term (c), then finding two numbers that multiply to this product and add to the coefficient of x (b). This method facilitates the factoring process by enabling a splitting of the middle term, which can then be grouped and factored further.

Transcript

00:01 All right, we're going to do this problem.

00:05 Factor trinomial.

00:07 Factor the trinomial.

00:10 Factor this trinomial.

00:13 And you can do this intuitively, or you can follow this general formula and the c, picking a, picking b's, picking c and picking d.

00:22 Because a and b, notice, a times b equals 3.

00:26 These are your constraints.

00:30 But they could be positive and negative.

00:32 B, my bad, a and c rather, i know my own variables.

00:42 A times c is 3 and b times d is going to be 5 in this case.

00:52 And b, these could be positive or negative, c could be positive or negative.

00:58 I mean, they both have to be positive or both negative.

01:02 But there are so many combinations you can try.

01:05 If you want to do this algorithmically, which is good the way to understand these kind of problems to make educated guesses in the future, we're going to skip forward and do educated guesses or intuitively solve these problems.

01:20 So factoring this, we're still going by the same logic of the constraints.

01:27 We know the answer has to look something like this.

01:31 And we have an x here to get the x squared for the front.

01:40 And we have a number at the back.

01:42 That wrote.

01:44 And these back numbers have to multiply to five.

01:48 And the front numbers, these ones, have to multiply to three.

01:53 So back, five, front three.

01:59 Okay.

02:05 Let's try.

02:06 We're going to put a one here.

02:09 Put a three.

02:12 And we're going to try positive.

02:19 Positive.

02:19 Let's see.

02:21 Positive 1 and positive 5 now the problem with this one is that what i was talking about it's not a good guess because we can't get negative 16 so we can't have but we have to have some negatives hope that makes sense because in this case we get 3x squared plus 3x let's go in order 5x 3x we get positive in the middle plus 5 that's not good so we have to have some negatives and it could be the front numbers or the back numbers so let's see where the x terms or the constants x terms these the constants are these so let's try let's try putting a negative here and putting negative here okay i think that will be better let's see so we still get 3x squared because both of them are negative not either of them in this case we have to have a both of them have to be a negative let's do let's do okay so next we get let me do this in order 5x because of this multiplication minus 3x okay so we're closer we got we got negative 8 x here but not quite negative 16 so let's erase that so raise these dots erase the negatives okay so maybe that will work we can try switching these the fives and the and the one that will give us a different answer we can also try doing negative 1 negative 5 without switching so let's try this first okay so we still get 3x squared now we get negative 5x we get negative 3x so we get the same thing now won't work so we'll get 5 at the end and the 3x square in the front but this won't do this won't do it's negative 8x versus negative 16x.

05:03 Okay, we can try switching these now because notice go back the one is being affected by the three and we want a greater number so maybe we want three multiplying by five so if we pair if we pair if we pair if we pair negative 5 here and negative 1 here with the 3 then we can have a multiplication going that way and that looks that looks that looks that we can get a bigger number and bigger negative number because 5 times 3 negative 5 times 3 is negative 15 right versus 3x times negative 1 which is 3 so hopefully that will get a 16 let's see maybe this will work 3x squared looks like it does negative x okay negative 1x times negative 1x the 1x times negative 1 also i put the 1x but that's not good notation just so you guys understand what's going on and you get negative 15x from the outers i mean uh inners foil foil i and and finally we get 5 at the end.

06:47 Okay, so it looks like this is a solution.

06:50 We get negative 16x in the middle.

06:53 And the 3x matches this, and the 5 matches this.

07:00 Now, i'm going to erase this.

07:03 Actually, let me put up a new one.

07:07 Problem is, now some of you, while testing, may get a different answer.

07:14 It looks something...

07:15 So the answer we got was 3x...