Factorise fully 𝑧 plus three squared minus 19 multiplied by 𝑧 plus three plus 90.
So, the first thing we can do before we can fully factorise our expression is distributing across our parentheses. And then, what we’re gonna do is simplify. And then, finally, what we’ll do is factorise what is left. So, as you see I’ve rewritten our expression as 𝑧 plus three multiplied by 𝑧 plus three, cause that’s what 𝑧 plus three squared is, minus 19 multiplied by 𝑧 plus three plus 90.
So, what we’re gonna do is first we’ll start with our double parentheses. So, we’ve got 𝑧 multiplied by 𝑧, which is 𝑧 squared. 𝑧 multiplied by positive three is positive three 𝑧. So, now, that’s the 𝑧 multiplied by both terms in the right-hand parentheses. So, we’re gonna do the same with the positive three. So, we first will get positive three 𝑧. And then, we add to it nine to cause we got three multiplied by three. Okay, great, now, let’s move on to the 19 multiplied by 𝑧 plus three.
So, first of all, we’re gonna have minus 19𝑧 and then minus 57, and we got that cause we had negative 19 multiplied by positive three, then plus 90. Okay, great, so, now, what we’re gonna do is simplify by collecting like terms. So, what we’re gonna get is 𝑧 squared. And then, we’ve got minus 13𝑧, and that’s cause we had three 𝑧 plus three 𝑧, which is six 𝑧, minus 19𝑧, which is negative 13 𝑧. So, then, what we’re gonna have is add 42. And we get this because we had positive nine. Then, we add 90, which is 99. Now, we subtract 57, which gives us 42.
So, now, we’re at the stage that we can factorise fully because we’ve got our quadratic. Our quadratic is 𝑧 squared minus 13𝑧 plus 42. Now, if we want to factorise this and put it into parentheses, what we need to do is find two factors whose product is positive 42 but whose sum is negative 13. Cause that’s our coefficient of 𝑧. So, first of all, we can take a look at some possible factors. So, we’ve got things that make 42. So, when they multiply together they make 42. So, the product is 42, six and seven, 21 and two, three and 14, and one and 42.
Well, the product of these two factors has to give us positive 42. So therefore, the numbers have to either both be positive, or both be negative. However, because the sum of both of our numbers, or factors, needs to be negative 13, that tells us that both of our factors does, in fact, have to be negative. So, now, what we need to do is see which of our pairs of factors when added together will give us a negative 13.
Well, when we add together our pairs of factors, we get negative six plus negative seven is negative 13. So, this is the one that we’re looking for cause the others gave us negative 23, negative 17, and negative 43, which aren’t the values that we’re looking for. So therefore, these are gonna be the factors in our parentheses. So, we can say that, fully factorised, the answer is 𝑧 minus seven multiplied by 𝑧 minus six. And it’s worth noting, at this point, it doesn’t matter which way round our parentheses go; either way would give us the correct answer.