Full transcript
0:00in this video we're going to talk about
0:02how to factor
0:03trinomials in this example we have a
0:06trinomial where the leading coefficient
0:08is 1.
0:09in order to factor it you need to find
0:11two numbers
0:12that multiply to the constant term 12
0:15but add to the middle coefficient 7.
0:18so what are those two numbers well let's
0:21begin by writing the factors of 12.
0:23if we divide 12 by 1 we get 12. if we
0:26divide 12 by 2 we get 6.
0:28if we divide twelve by three we get four
0:31now which of these pairs of numbers
0:33multiply to 12 but add to seven we know
0:36it's three and four
0:37three plus four is seven but three times
0:40four is twelve
0:41so to factor the trinomial we can write
0:43it as x plus 3
0:45times x plus 4. and so that's how we
0:49could factor
0:50that particular trinomial now let's
0:53consider another example
0:54so let's say we have x squared
0:58plus 11x
1:03plus 30. go ahead and factor that
1:07particular trinomial
1:10feel free to pause the video if you want
1:11to
1:14so first we need to find two numbers
1:16that multiply to 30
1:18but add to the middle coefficient 11. so
1:21let's write down the factors of 30.
1:23dividing by 30 i mean if we divide 30 by
1:251 rather
1:26we get 30. if we divide it by 2 we get
1:2815.
1:2930 divided by 3 is 10. now we can't
1:32divide 30 by 4 evenly
1:344 doesn't go into 30. if you do 30
1:36divided by 4 you're going to get 7.5
1:39but we could divide it by five and get
1:40six and then if we continue further if
1:43we divide it by six we get five
1:46once the number is reversed you can stop
1:47it there
1:49now which pair of numbers multiply to 30
1:52but add to 11.
1:54we know that 5 plus 6 add to 11.
1:57so the answer is going to be x plus 5
2:01times x plus 6. so that's how we can
2:03factor
2:04x squared plus 11x plus 30. that's what
2:07you can do if the leading coefficient
2:09is equal to one now let's work on
2:13another example
2:14this time we're going to solve a
2:17quadratic equation
2:19so let's say we have the equation x
2:21squared minus 5x
2:22plus 6 equal to 0.
2:26how can we calculate the value of x that
2:28will make this equation true
2:31well since we have a trinomial on the
2:33left side we can factor
2:35so we need to find two numbers that
2:37multiply to six but add up to negative
2:39five
2:40if we divide six by one we get six if we
2:42divide it by two we get three
2:45now two plus three add up to five but if
2:48we add two negative signs it will add to
2:50negative five
2:51and negative two times negative three is
2:53still equal to positive six
2:55so to factor it's going to be x minus
2:57two
2:58times x minus three so now that we've
3:02factored the trinomial we can now solve
3:04the equation
3:05using the zero product property we can
3:08set each factor
3:09equal to zero for the first one if we
3:13add two to both sides
3:14we'll get that x is equal to two for the
3:17second one if we add three to both sides
3:20we'll get that x is equal to three
3:25and so these are the two solutions to
3:27this
3:28quadratic equation try this example
3:31let's say that x squared plus
3:353x
3:39minus 28 is equal to zero
3:43go ahead and solve for this particular
3:45quadratic equation
3:48so let's begin by factoring it let's
3:50find two numbers that multiply to
3:52negative 28
3:53but add to positive three so
3:56if we divide 28 by one we're going to
3:58get negative 28.
4:00if we divide it by 2 we'll get negative
4:0214. 3 doesn't go into
4:0428 but we can divide it by 4 and get
4:06negative 7.
4:08and then if we divided by 7 we'll get
4:10negative 4
4:11and we'll stop there now notice that
4:15four plus negative seven adds up to
4:17negative three
4:19however positive seven plus negative
4:22four
4:22adds up to positive three and this is
4:25what we wanna use
4:27so when factoring it it's going to be x
4:29plus seven
4:30times x minus four
4:36now let's set each factor equal to zero
4:41let's go ahead and solve this this
4:42equation so first let's subtract both
4:45sides by seven
4:46and this will give us the first answer x
4:48is equal to negative seven
4:51and for the second one if we add four to
4:52both sides we get x is equal to four
4:56so we have our two possible solutions x
4:59is equal to negative seven
5:00or x is equal to four and what you could
5:03do to check your work is you can plug it
5:04back in
5:05to the original equation let's plug in
5:08four
5:11four squared plus three times four minus
5:1528 let's see if that's going to give us
5:16zero
5:17four squared or four times four is
5:19sixteen three times four is twelve
5:22sixteen plus 12 is 28 28 minus 28 is
5:26zero
5:27so we could see that this indeed works
5:30now let's work on an example where the
5:32leading coefficient is not equal to one
5:35in this example the leading coefficient
5:37is 2.
5:40how can we solve this particular
5:42quadratic equation
5:46how can we factor the trinomial
5:50well to do that what you need to do is
5:52multiply the leading coefficient by the
5:54constant term
5:562 times 6 is equal to 12.
5:59next we need to find two numbers that
6:01multiply to 12 but add to the middle
6:03coefficient
6:04negative 7. so let's write down the
6:06factors of twelve
6:07it's one and twelve two and six three
6:10and four
6:12now three plus four add up to positive
6:14seven
6:15but if we use negative three and
6:17negative four it will add up to negative
6:197
6:19and still multiply to positive 12.
6:24now for this type of problem we can't
6:26just write the answer immediately
6:28there's going to be some additional
6:29steps here what we're going to do now
6:31that's different
6:32is we're going to replace the middle
6:35term
6:36with these two
6:39so you can write negative 3x minus 4x or
6:41you can write negative 4x minus 3x
6:44i'm going to write it this way negative
6:464x minus 3x
6:49the reason being is these two are even
6:51numbers and you can take out a two
6:54and for those two numbers they're
6:55multiples of three you can easily factor
6:57out a three
6:59now let's factor by grouping in the
7:02first two terms we're going to take out
7:04the gcf
7:05which is 2x if you divide 2x squared by
7:082x
7:09you're going to get x if you divide
7:11negative 4x by 2x
7:13you're going to get negative 2. now in
7:16the next two terms we're going to take
7:17out the gcf or the greatest common
7:19factor
7:20which is going to be negative three
7:22negative three x divided by negative
7:24three is x
7:25positive six divided by negative three
7:28is negative two
7:31now if these two are the same you know
7:33you know that you're on the right track
7:35so we're gonna factor the gcf out of
7:38these two terms and the gcf is what i
7:41have highlighted in blue
7:42it's x minus two so if you take out x
7:45minus two from that term
7:47you're left with two x if you take out x
7:50minus two from this term
7:51you're left with minus three so this is
7:55how you can factor a trinomial
7:57where the leading coefficient is not one
8:00now to check our work what we could do
8:02is foil
8:04if we foil these two binomials we should
8:06end up with the trinomial that we
8:08started with
8:09a binomial is basically a polynomial
8:12expression with two terms
8:13a trinomial is one with three terms one
8:16two
8:16three i had four marks but this is one
8:20two three if we multiply
8:24x by two x we're going to get 2x squared
8:27if we multiply x by negative 3 we're
8:30going to get negative 3x
8:32negative 2 times 2x is negative 4x
8:35negative 2 times negative 3 is plus 6.
8:39and then these two here will add up to
8:42negative seven
8:42x as you can see these two expressions
8:46are equivalent
8:47so that tells us that we are on the
8:50right track
8:53now the last thing we need to do is
8:55solve the equation
8:58so let's set each factor equal to zero
9:00using the zero product property
9:05so if we add two to both sides for the
9:07first equation
9:08we can get the first solution which is x
9:10is equal to 2.
9:12for the second one if we add 3 to both
9:14sides
9:16we'll get that 2x is equal to 3 and then
9:19we'll have to divide both sides by two
9:22so we get the solution x is equal to
9:25three over two
9:27so here are the two answers x is equal
9:30to two
9:31and x is equal to three over two
9:35now for the sake of practice let's work
9:37on another example
9:38and for this one we're not going to try
9:39to solve the equation because you
9:41already know how to do that at this
9:42point
9:43we're simply going to focus on factoring
9:45the trinomial
9:47well since the leading coefficient is
9:49not 1 it's 3 this time
9:51we're going to multiply the leading
9:52coefficient by the constant term
9:56so 3 times negative 6 is negative 18.
10:01and then we're going to look for two
10:03numbers that multiply it to negative 18
10:05but add to the coefficient the middle
10:07coefficient of negative 7.
10:10so let's write down the factors of
10:11negative 18.
10:14if we divided by 2 we're going to get
10:15negative 9. if we
10:17it by 3 we'll get negative 6 and if we
10:19divide it by 6 we'll get negative 3.
10:23now note that 2 and negative 9 they
10:26multiply to negative 18
10:27and they add up to the middle
10:29coefficient negative 7
10:31which is what we want
10:34so we can replace negative 7x with
10:37positive 2x and negative 9x or we can
10:40reverse it
10:41and write it as negative 9x plus 2x
10:46in either case these two they still add
10:48up to negative 7x
10:51but i chose it to i chose to put it that
10:53way because
10:55when factoring by grouping it's just
10:57going to be a lot easier
10:59in my opinion
11:02now let's take out the gcf in the first
11:04two terms
11:06so that's going to be 3x 3x squared
11:09divided by 3x
11:10is x negative 9x divided by 3x
11:15is negative 3. now for the last two
11:18terms let's take out the gcf which is
11:20going to be 2.
11:222x divided by 2 that's
11:26x negative six divided by two is
11:29negative three
11:31next let's factor out x minus three
11:36so if we take out x minus three in the
11:38first term we're gonna have three x
11:41and then if we take it away in the
11:42second term we're going to have 2.
11:45so this is the answer x minus 3
11:49times 3x plus 2. and just to check it
11:53x times 3x is 3x squared x times two
11:56is two x negative three times three x is
12:00negative nine x
12:01negative three times two is negative six
12:04and then two x minus nine x is negative
12:06seven x
12:08so we start with the original expression
12:17so this right here is the answer
12:23now let's try a challenging problem
12:27so consider the trinomial 72 x squared
12:31plus 17x
12:38minus 70. how would you factor
12:41this particular trinomial well
12:45if you follow the steps that we've been
12:47using
12:48you know the first thing to do is to
12:49multiply the leading coefficient by the
12:51constant
12:52term 72 times
12:55negative 70 that's negative 5040.
13:00now it can be challenging to find two
13:02numbers that
13:03multiply to 5040 or negative 5040
13:07and at the same time add up to 17.
13:12now the best way to do this is to do it
13:14systematically
13:15and find a just go by the list let's say
13:19if you divide this by 1 you get
13:21negative 50 40. if you divide it by 2
13:25you get negative 25 20.
13:28and then if you keep doing this
13:29eventually you're going to find two
13:30numbers
13:32that will multiply it to negative 50 40
13:34and add to 17.
13:36but that's going to take a lot of time
13:38so is there a more efficient way
13:41in which we can factor this trinomial it
13:44turns out that there is
13:49and it involves using the quadratic
13:52formula
13:52let's set this equal to zero
13:56let's begin by writing the quadratic
13:58formula
13:59it's equal to negative b plus or minus
14:02the square root of b squared minus 4ac
14:07divided by 2a a
14:10is the leading coefficient 72 b is the
14:13number in front of x
14:14c is the constant term so in this
14:16example
14:18we have negative 17 plus or minus the
14:21square root
14:23of 17 squared minus 4 times
14:27a times c
14:31divided by 2a or 2 times 72
14:35so let's simplify it
14:4117 squared is 289
14:45and if we multiply negative 4 by 72
14:49times negative 70 that's going to be
14:5220 160 and 2 times 72
14:57is 144.
15:01now 289 plus 20160
15:04that's 20 449. if you take the square
15:07root of that number
15:09you're going to get 143.
15:13now we can break this down into two
15:15answers the first one
15:17is going to be keep in mind we have two
15:21signs
15:22so the first one is going to be negative
15:2417
15:25plus 143 divided by
15:28144
15:32negative 17 plus 143 that's 126.
15:37now we need to reduce this fraction
15:41we could divide both numbers by 2. 126
15:44divided by 2 is 63
15:46144 divided by 2 is 72
15:50now 63 is 9 times 7 seventy-two is nine
15:54times eight
15:55so this reduces to seven over eight
15:59so this is one of the answers to the
16:01equation x
16:02is seven over eight now let's get the
16:06other answer
16:10so instead of using the plus sign we're
16:12now going to use the minus sign
16:14negative 17 minus 143
16:18over 144
16:22negative 17 minus 143
16:28that is negative 160.
16:31now let's see how we can break this into
16:34smaller numbers
16:35if we divide by 2 half of negative 160
16:38is negative 80
16:40half of 144 is 72. so these are
16:43numbers that we can deal with so we know
16:45that 72 is 9 times 8.
16:48and 80 is 10 times 8 and let's not
16:52forget the negative sign
16:53so if we cancel an 8 we're left with
16:57negative 10 over 9. so that's the second
17:01solution to this equation
17:04now remember our goal is not necessarily
17:07to solve the quadratic equation
17:08because we have those answers but our
17:11goal is to find
17:12an easy way to factor this expression
17:15and basically we're reverse factoring it
17:17using the quadratic formula so here's
17:20what we're going to do now
17:22we're going to write each solution
17:28now from this point we need to get a 0
17:30on one side of the equation
17:33to do that i'm going to multiply both
17:35sides by eight
17:37for the first equation here the eights
17:39will cancel and i get eight x
17:41is equal to seven next i'm going to
17:44subtract both sides by seven
17:48so on the left i'm going to get eight x
17:50minus seven and on the right is zero
17:53you can't really combine these terms
17:54because they're
17:56they are unlike terms now let's do the
17:58same thing for the other
18:00solution let's multiply both sides by 9.
18:03this will give us 9x is equal to
18:05negative 10. now i'm going to move the
18:07negative 10 from the left from the right
18:08side
18:09to the left side as i move it from the
18:12right side to the left side is going to
18:13change side
18:14it's going to go from negative 10 to
18:15positive 10. and so i'm going to get 9x
18:18plus 10
18:20is equal to 0. so
18:23these two things right here are the
18:26factors to the trinomial
18:30so thus to factor the trinomial we can
18:32simply write
18:34eight x minus seven times nine x
18:38plus ten
18:41and so that's how we can reverse factor
18:44the trinomial
18:45using the quadratic formula
18:49so this right here is the answer and if
18:52we want to check it
18:53just to make sure we have it right it's
18:56going to be 8x
18:57times 9x which is 72x squared
19:00and then 8x times 10 that's 80x
19:05negative 7 times 9x that's negative 63x
19:10negative 7 times 10 is negative 70. and
19:13then if we combine the two middle terms
19:1680 minus 63 is 17.
19:21and so we get the original expression
19:23that we started with
19:25so now you know how to reverse factor a
19:28trinomial
19:29using the quadratic formula