Multiplying monomials by polynomials. So we can see that this is a monomial as we have three. That’s a monomial, meaning basically one term. And we’re multiplying it by a polynomial inside the parentheses which means basically more than one term. Well what we’re gonna do for this is apply the distributive property, where 𝑎 multiplied by all of 𝑏 plus 𝑐 is equal to 𝑎𝑏 plus 𝑎𝑐. So what we mean by that is we gonna take three or take 𝑎 and multiply it by the first term. So in this case will do three multiplied by two 𝑥. And then we’ll add to that three multiplied by the second term. So three multiplied by negative seven. And now we need to simplify it. So we know that three multiplied by two is six and then we can put an 𝑥 after it, cause there is an 𝑥 there as well. So six 𝑥 and then three multiplied by negative seven is negative twenty-one. So nice and simply, all we did there is we took a monomial and multiply it by the first term in the parentheses and then we added that to the multiplication of our monomial by the second term of the parentheses.
Let’s have a look at a different example. So we can see here with this next example, we’ve got a couple of things that are different. First of all, a monomial on the outside of the parentheses is not just a constant. It’s got a variable and also is negative. The other thing we can see is inside the parentheses we have three terms. Well it doesn’t really matter to us, because we’re going to use exactly the same method to expand these parentheses. We’re gonna use the distributive property. So first thing first good place to start, we’ll take the monomial and we’ll multiply it by our first term. And then we’ll add on to that monomial negative five 𝑥 multiplied by our second term. Be careful! Our second term don’t forget the negative in front of the two, because it’s always the sign in front of it that’s attached. So we’ve got negative five 𝑥 multiplied by negative two 𝑥. And then we need to multiply the last term, so we’ll have negative five 𝑥 multiplied by negative eight.
So then let’s try and simplify. We’ve got negative five. We multiply it by three. We get negative fifteen and then we know that 𝑥 multiplied by 𝑥 squared is the same as 𝑥 multiplied by 𝑥 multiplied by 𝑥. So we’ve got negative fifteen 𝑥 to the power of three or 𝑥 cubed. Then look at the signs for the next one. So we’ve got two negatives multiplied together. So that will cancel out. So we’ll have a positive. Now we’ve got five multiplied by two which is ten, and then 𝑥 multiplied by 𝑥 which is 𝑥 squared. So we’re adding on ten 𝑥 squared.
Now again let’s look at the signs. First, we’ve got a negative multiplied by a negative which gives us a positive. Then we’ve got five multiplied by eight which is forty. Then we’ve got an 𝑥, so add on forty 𝑥. So there we have it. We have completely multiplied out this set of parentheses. We’ve multiplied this monomial by this polynomial. For our next one, let’s have a look when it’s not just 𝑥 but also 𝑦.
So now we have five 𝑥 squared all multiplied by three 𝑦 plus two 𝑥 plus 𝑥𝑦. So again, we can see that we’ve got some 𝑥s and some 𝑦s, but we could have any variables. We’re gonna do exactly the same thing either way. We’re gonna take our first term and we’re gonna multiply it by a monomial. So we’ve got five 𝑥 squared multiplied by three 𝑦. And then we’re gonna add on to that monomial by the second term, so five 𝑥 squared multiplied by two 𝑥. And then five 𝑥 squared multiplied by 𝑥𝑦.
And we’re just gonna take it one term at a time. So we’ll do the numbers first. We’ve got five multiplied by three. We know that’s fifteen. And then 𝑥 squared multiplied by 𝑦, so it’s fifteen 𝑥 squared 𝑦. Then for our next one, it’s five multiplied by two which we know is ten. And then looking at the 𝑥s, we’ll have 𝑥 squared which is 𝑥 times 𝑥, then we’re timesing that by another 𝑥. So that gives us ten 𝑥 cubed or ten 𝑥 to the power of three. Then for the last one we’ve just got five, because it’s five times one. And then looking at the 𝑥 powers, we’ve got five 𝑥 to the power of three and then 𝑦. So there we have it. We’ve done that one as well. All we’ve done is we take the monomial and multiplied it by each term individually. Let’s have a look at our final example.
So before we do anything for this question, I want you to just have a look at it and think what’s gonna be the very first step to multiply out this monomial. And now I hope that none of you have thought two plus three, because it’s not two plus three then multiplied by the rest. It’s two plus three multiplied by everything. So the two is just by itself to add at the end. So what we’re gonna do and what you should always do in these things is first of all write the thing by itself straight down, so you don’t get tempted to do anything with it.
And now we’ve got a nice and simple multiplication to get on with and then collect the light terms after. So we’ve got three multiplied by the first term: so three multiplied by 𝑥 squared then three multiplied by two 𝑥 and then three multiplied by negative seven. So taking it one term at a time, we’d just write two again straight down, not to do anything with it. And we’ll add three multiplied by 𝑥 squared which is three 𝑥 squared. Three multiplied by two 𝑥, well three times two is six, so we’ve got six 𝑥. And then three multiplied by negative seven, well three multiplied by seven is twenty-one. Check the negative in front of it, because it’s a positive and a negative gives us a negative. So we’ve got negative twenty-one.
In this case, we’re not actually finished yet and that is because we have got some like terms that we need to collect. So what we’re gonna do is we’ve got negative twenty-one, add two. So we’ll have three 𝑥 squared plus six 𝑥 minus nineteen. And there we have it. We finished it. So be careful that whenever you’ve got something plus a monomial multiplied by a polynomial, it’s there to trick you. Make sure that you do not add it first, or in the case where it was, say two 𝑥 plus three all multiplied by 𝑥 squared plus two 𝑥 minus seven, that you don’t even worse try and make it a binomial multiplied by that polynomial. Because that is a much more challenging thing to work out and you’ll end up wasting loads of time doing something that’s not going to get you any marks. So be careful. Pay attention to parentheses to make sure that you know exactly what you’re doing.