Video Transcript
Factorize fully π₯ plus four π¦ plus three squared minus π₯ minus four π¦ minus three squared.
The key here is to notice that one square term is being subtracted from another square term. And we call this the difference of squares. π squared minus π squared is equal to π plus π times π minus π. We can use this to help us factorize.
For us, π equals π₯ plus four π¦ plus three. And π equals π₯ minus four π¦ minus three. We need π plus π times π minus π, which would look like this. The key here is to be really careful with our positive and negative signs, π₯ plus four π¦ plus three plus π₯ minus four π¦ minus three and then π₯ plus four π¦ plus three. This time, weβre subtracting. So we need to distribute the subtraction. Minus π₯ minus negative four π¦ equals plus four π¦. And minus negative three equals plus three.
The next step will be to combine like terms. π₯ plus π₯ equals two π₯. Positive four π¦ minus four π¦ cancels out. Positive three minus three also cancels out. And that means our π plus π term has been simplified to two π₯.
On the other side, we have the π minus π terms. Positive π₯ minus π₯ cancels out. Four π¦ plus four π¦ equals eight π¦. And three plus three is six. Our π minus π terms equals eight π¦ plus six. But eight π¦ and six share a common factor. They both have a factor of two. If we take out that common factor of two, we could rewrite this to say, two times four π¦ plus three.
And remember, weβre multiplying the π plus π times the π minus π. We would then have two π₯ times two times four π¦ plus three. We can multiply this, two times two, which equals four. Bring down the π₯. And then bring down the four π¦ plus three.
The fully factorized form is four π₯ times four π¦ plus three.
