WEBVTT
00:00:00.810 --> 00:00:07.910
Factorize fully five π¦ to the fourth plus 40π§ squared plus 30π¦ squared π§.
00:00:08.950 --> 00:00:10.140
Hereβs our expression.
00:00:10.930 --> 00:00:16.060
The first thing we notice is that all three coefficients are divisible by five.
00:00:16.840 --> 00:00:19.280
So we want to take out a factor of five.
00:00:19.930 --> 00:00:22.300
When we do that, weβre left with π¦ squared [π¦ to the fourth].
00:00:22.710 --> 00:00:25.850
40 divided by five equals eight.
00:00:26.310 --> 00:00:28.190
And the variable doesnβt change.
00:00:28.570 --> 00:00:31.540
30 divided by five equals six.
00:00:31.780 --> 00:00:33.460
And the variable doesnβt change.
00:00:34.330 --> 00:00:38.130
Notice how our leading term is π¦ to the fourth.
00:00:38.840 --> 00:00:43.490
We see that we have another term that has a π¦-variable, a π¦ squared.
00:00:43.760 --> 00:00:48.610
And we wanna change the order so that π¦ squared term comes next.
00:00:49.340 --> 00:00:50.510
Itβd look like this.
00:00:50.720 --> 00:00:56.610
Five times π¦ to the fourth plus six π¦ squared π§ plus eight π§ squared.
00:00:57.530 --> 00:01:03.300
Once we do that, we can see that we can factor this expression into two smaller expressions.
00:01:04.340 --> 00:01:08.800
Our π¦ to the fourth is found by multiplying π¦ squared by π¦ squared.
00:01:09.450 --> 00:01:13.270
And our π§ squared would be found by multiplying π§ times π§.
00:01:13.870 --> 00:01:16.480
We canβt forget this coefficient of eight.
00:01:17.220 --> 00:01:23.560
We need two factors that multiply together to equal eight and add together to equal six.
00:01:23.980 --> 00:01:27.740
The factors of eight are one and eight and two and four.
00:01:28.400 --> 00:01:31.080
If we add two plus four, we get six.
00:01:31.700 --> 00:01:42.210
And that means we want to use this two and four as coefficients of our π§-variable, π¦ squared plus two π§ times π¦ squared plus four π§.
00:01:42.710 --> 00:01:44.930
And we canβt forget to bring down the five.
00:01:45.560 --> 00:01:56.480
At this point, itβs probably good to go back and check these two expressions to make sure we end up with what we started with, π¦ squared plus two π§ times π¦ squared plus four π§.
00:01:56.700 --> 00:02:00.090
π¦ squared times π¦ squared equals four π¦ [π¦ to the fourth].
00:02:00.530 --> 00:02:05.020
π¦ squared times four π§ equals four π¦ squared π§.
00:02:05.370 --> 00:02:09.780
Two π§ times π¦ squared equals two π¦ squared π§.
00:02:10.110 --> 00:02:22.870
Now we have two π¦ squared π§ plus four π¦ squared π§, which equals six π¦ squared π§, and then two π§ times four π§, which equals eight π§ squared.
00:02:23.400 --> 00:02:25.800
Weβre multiplying all of this by five.
00:02:26.350 --> 00:02:30.090
Five times π¦ to the fourth equals five π¦ to the fourth.
00:02:30.670 --> 00:02:35.500
Five times six π¦ squared π§ equals 30π¦ squared π§.
00:02:36.180 --> 00:02:43.290
Five times eight π§ squared equals 40π§ squared, which is what we originally started with.
00:02:44.020 --> 00:02:50.760
The fully factorised form is five times π¦ squared plus two π§ times π¦ squared plus four π§.