Expand and fully simplify 𝑥 minus two multiplied by 𝑥 minus one.
So in this question, we’ve been given a pair of brackets and asked to do two things. Firstly, we need to expand the brackets. And then, we need to fully simplify the result. To expand a pair of brackets means to multiply them out. And we need to make sure that we multiply each term in the first bracket, so that’s the 𝑥 and the negative two, by each term in the second bracket, that’s the second 𝑥 and the negative one.
Now, there’s an acronym that we can use to make sure that we multiply all the correct pairs of terms together. And that acronym is FOIL. An acronym is just a word made up of the first letters of other words. The F in FOIL stands for first. So this means we need to multiply the first term in each bracket together. We have 𝑥 multiplied by 𝑥 which is equal to 𝑥 squared.
Next, we have the O in FOIL which stands for outers. This means the terms on the outside of the brackets. So that’s the 𝑥 in the first bracket and the negative one in the second. We have 𝑥 multiplied by negative one which is equal to negative one 𝑥. But remember, we don’t need to write the one. So we just write this as negative 𝑥.
The I in FOIL stands for inners. So those are the terms on the inside of the expression. That’s the negative two and the 𝑥. We have negative two multiplied by 𝑥 which is equal to negative two 𝑥.
Finally, we have L which stands for lasts. This means we need to multiply the last term in each bracket together. So this is negative two multiplied by negative one. This gives positive two. And to see this, we need to remember the rule that a negative number multiplied by another negative number gives a positive answer.
So we’ve now completed the first stage of what we’re asked to do. We’ve expanded the pair of brackets. And it’s given 𝑥 squared minus 𝑥 minus two 𝑥 plus two. Notice that there’re four terms at this stage. And that will always be the case if you expand the pair of brackets in which each bracket has two terms.
The second part of what we’re asked to do is to fully simplify our answer. And this means that we need to have a look for any like terms, and then collect them together. In the centre of our expansion, we have negative 𝑥, or negative one 𝑥, minus two 𝑥 which overall makes negative three 𝑥. So we can collect these terms together. There are no further like terms. So we fully simplified our expansion. Our answer is 𝑥 squared minus three 𝑥 plus two.