Determine whether the following expression is prime or not: two 𝑥 squared minus seven 𝑥 minus four. If it is not prime, give the expression in its factorized form.
Well, if an expression is prime, what it means is that it cannot be factorized any further. So let’s work out whether our expression is in fact a prime. Well, to help us decide whether our expression is prime or not, what we can use is something called the discriminant. Well, what the discriminant is something that’s really useful. So if we’ve got a quadratic in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, then the discriminant is 𝑏 squared minus four 𝑎𝑐. But how’s this gonna be useful for our problem? Well, what we know is that if the discriminant 𝑏 squared minus four 𝑎𝑐 is greater than zero, then there are two real roots.
So therefore, we know that our expression would not be prime. And we also note if 𝑏 squared minus four 𝑎𝑐 was equal to zero, then we’ve got a repeated root. So again, our expression would not be prime. And what we also know is that if 𝑏 squared minus four 𝑎𝑐 is less than zero, then there are no real roots. And this is when we know that our expression is gonna be prime because it cannot be factorized. Great, so now we know what the discriminant is and we know how to use it. Let’s try using it with our expression. Well, 𝑎 is equal to two, 𝑏 is equal to negative seven, and 𝑐 is equal to negative four in our expression. So that’s our 𝑎, 𝑏, and 𝑐 because our 𝑎 is the coefficient of 𝑥 squared, 𝑏 is the coefficient of 𝑥, and then we’ve got 𝑐, which is our numerical value on the end.
Well then, therefore, if we substitute our values in, we’re gonna get 𝑏 squared minus four 𝑎𝑐. So our discriminant is equal to negative seven squared minus four multiplied by two multiplied by negative four which is gonna be equal to 49 plus 32. And this could be four multiplied by two multiplied by negative four is negative 32. And if we subtract a negative, it’s the same as adding. So this is gonna be equal to 81. So therefore, we can say that definitely our expression is not prime. So we can factorize it. So now what we want to do is factor or factorize our quadratic. And to do that, what we need to do is put it into two pairs of parentheses. And we could do that with some trial and error. However, there is a method that we can use if you want to, which makes things easier when we’ve got coefficient of 𝑥 squared greater than one.
Well, first of all what you do is multiply 𝑎 and 𝑐. So in this case, that’s gonna be two multiplied by negative four which is gonna give the answer negative eight. And then what we want to find is two factors whose product is negative eight which we just found and whose sum is the coefficient of 𝑥, which is negative seven. Well, our two factors are gonna be negative eight and positive one. That’s because negative eight multiplied by positive one is equal to negative eight and negative eight add one is equal to negative seven.
Okay, great, so now what’s the next step? So now, what we do is we split up our coefficient of 𝑥 into the two factors we got, which was negative eight and positive one. So we’re gonna have two 𝑥 squared minus eight 𝑥 plus one 𝑥 minus four. You wouldn’t usually write the one 𝑥 but I’m putting the one in front of the 𝑥 just cause it’s gonna help with the next stage. We can see what’s happening. So now, what we need to do is factor or factorize the first two terms together and then the second two terms. Well, if we factorize the first two terms, we’re gonna take two 𝑥 out as a factor. And then inside the parentheses, we’ll have 𝑥 minus four. And that’s because two 𝑥 multiplied by 𝑥 is two 𝑥 squared and two 𝑥 multiplied by negative four is negative eight 𝑥. And then we take out one as a factor of our last two terms. So what we’re gonna have is plus one then multiplied by 𝑥 minus four.
So now, to form our two factors, what we do is take the first part before both of the parentheses. So we got two 𝑥 and then add one. And then for a second pair of parentheses what we’re going to take is the value inside the parentheses in this line, so 𝑥 minus four. It’s also a good check because we know that we’re on the right track because we have the exact same one in each of our parentheses 𝑥 minus four. If these were different, then you’d have done something wrong. Then you need to check. So therefore, we’re gonna get two 𝑥 plus one multiplied by 𝑥 minus four.
So therefore, we’ve solved the problem cause we’ve shown that the expression is not prime. And we’ve shown that its fully factorized form is two 𝑥 plus one multiplied by 𝑥 minus four. But what we can do now quickly is check this by distributing across the parentheses. So if you have two 𝑥 multiplied by 𝑥, it’s two 𝑥 squared. Two 𝑥 multiplied by negative four gives us negative eight 𝑥. One multiplied by 𝑥 gives us plus 𝑥. And then finally, positive one multiplied by negative four gives us negative four. So then if we simplify, we get two 𝑥 squared minus seven 𝑥 minus four which is what we started with. So we’ve doubled checked and we definitely have the correct answer.