Video: Pack 4 • Paper 2 • Question 11

Pack 4 • Paper 2 • Question 11

04:09

Video Transcript

Expand and fully simplify the following expression: two 𝑥 plus two multiplied by five 𝑥 minus three multiplied by 𝑥 plus eight.

Now in order to actually solve this problem, what we’re actually gonna do is expand pairs of brackets at a time. But the first thing we’ve actually done is taking a two as a factor outside the first bracket. And the reason we’ve done this is actually because it’s gonna make our calculations a little easier. You don’t need to do this step. If you’re confident actually just multiplying brackets, you can do that. We’ve done that just to make it a little bit simpler for this question.

So for the first stage, what we’re actually gonna do is expand the first two brackets. What I’m gonna do is actually show you how to do that just to remind you. So first of all, what we do is we multiply the first term in our first bracket with the first term in the second bracket. So that’s 𝑥 multiplied by five 𝑥, which gives us five 𝑥 squared. And then what we do is we actually multiply the 𝑥 by the other term in the second bracket, which is negative three. So we get negative three multiplied by 𝑥, which gives us negative three 𝑥.

So then what we do is we actually move on to the second term in the first bracket, because we can see that the first term has been multiplied by every term in the second bracket already. So we’ve got positive one multiplied by five 𝑥, which gives us positive five 𝑥. And then finally we have positive one multiplied by negative three, which gives us negative three.

So therefore, we can actually put that back into our expression. So what we have is two multiplied by five 𝑥 squared minus three 𝑥 plus five 𝑥 minus three multiplied by 𝑥 plus eight.

Okay, so the next stage now is to actually simplify this. And we do that by combining our like terms. So we have negative three 𝑥 plus five 𝑥, which gives us two 𝑥. So we’re now left with two multiplied by five 𝑥 squared plus two 𝑥 minus three multiplied by 𝑥 plus eight.

So now what we’re gonna do is actually expand the next pair of brackets. And we do this in exactly the same way that we’ll expand any brackets. It doesn’t matter that the first bracket contains three terms. I’m gonna show you how we do that now.

So we’re gonna multiply five 𝑥 squared by 𝑥, which is gonna give us five 𝑥 cubed. Then what we’re gonna do is actually multiply this term by the other term in the second bracket. So that’s going to be five 𝑥 squared multiplied by positive eight, which gives us positive 40𝑥 squared.

So now what we do is we actually move on to the next term in the first bracket. So it’s positive two 𝑥. So we’ve got positive two 𝑥 multiplied by 𝑥, which gives us positive two 𝑥 squared. And then we have positive two 𝑥 multiplied by positive eight, which will give us positive 16𝑥.

And now we can actually move to the final term in the first bracket. So we start with negative three multiplied by 𝑥, which gives us negative three 𝑥, and then finally negative three multiplied by positive eight, which gives us negative 24.

Okay, great! So we’ve expanded those brackets. We can actually put this back into our expression. So therefore, what we’ve got is two multiplied by five 𝑥 cubed plus 40𝑥 squared plus two 𝑥 squared plus 16𝑥 minus three 𝑥 minus 24.

So what we’re going to do again here is actually collect like terms and simplify. So we get two multiplied by five 𝑥 cubed. And then we’ve got positive 40𝑥 squared plus two 𝑥 squared, which is gonna give us positive 42𝑥 squared. And then we have positive 16𝑥 minus three 𝑥, which gives us positive 13𝑥, and then our final term which is minus 24.

Okay, great! So now at this stage, we’ve got our expression to two multiplied by five 𝑥 cubed plus 42𝑥 squared plus 13𝑥 minus 24, one more stage to go. So the final stage is to actually multiply through by that factor of two that we took out earlier. So we get two multiplied by five 𝑥 cubed gives us 10𝑥 cubed. Two multiplied by 42𝑥 squared gives us 84𝑥 squared. Two multiplied by 13𝑥 gives us 26𝑥. And two multiplied by negative 24 gives us negative 48.

So therefore, we can say that if we fully simplify the expression two 𝑥 plus two multiplied by five 𝑥 minus three multiplied by 𝑥 plus eight, we’re gonna get 10𝑥 cubed plus 84𝑥 squared plus 26𝑥 minus 48.

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