Completely factor six 𝑥 squared minus 19𝑥 plus 10.
Well this is a quadratic expression in 𝑥. We’ve got some 𝑥 squared; we’ve got some 𝑥 ; and we’ve got a number on its own. So we’re gonna be factoring this into a pair of parentheses like this; we’ve got some
𝑥s; and we’ve got some 𝑥s; and we’ve got some numbers.
Now this is what we call a nonmonic quadratic because the number of 𝑥 squareds is
not one. If it was only one 𝑥 squared, that would be a monic quadratic. And that makes our job a little bit more difficult, so let’s have a look at how we
can break this down and factor this expression.
Well first of all, let’s take the coefficient of 𝑥 squared and the constant term on
the end and multiply them together. Well, we’ve got positive six and positive ten. If we multiply those two together we get 60.
Now let’s just list out all the factors of 60. Well obviously one times 60 is 60, but two times 30 is also 60, three times 20, four
times 15, five times 12, six times 10.
Seven, eight, and nine don’t go into 60 exactly, and then we’re up to 10. And we’ve already encountered 10 in our list of factors here. So there are all the factors. And don’t forget, we’ve got the negative versions; negative one times negative 60 is
also positive 60.
Now we need to look through our list and find the pair of factors that add up to the
coefficient of 𝑥 there, negative 19. So which pair of factors here when you add those numbers together add up to negative
19? Well negative four add negative 15 make negative 19, so these are the pair of factors
I’m looking for.
So I’m gonna rewrite the negative 19𝑥 that we had in original expression there as
negative four 𝑥 take away 15𝑥. Now if I evaluate this second line, I can find negative four 𝑥 take away 15𝑥 is
still negative 19𝑥, so that line is still equal to the line above it.
But now I’m gonna consider it as two different expressions added together. So I’m gonna take the first half and the second half separately and factor each
one. So looking at the first one first, let’s fully factor six 𝑥 squared minus four
𝑥. Well the highest common factor of six and four is two, and the highest common factor
of 𝑥 squared and 𝑥 is 𝑥.
So those first two terms there factor to two 𝑥 times three 𝑥 because two 𝑥 times
three 𝑥 is six 𝑥 squared and two 𝑥 times negative two because two 𝑥 times
negative two is negative four 𝑥.
Now I’m going to copy down that set of parentheses here and work out what would I
need to multiply that by in order to generate those second two terms here. What do I need to multiply three 𝑥 by to get negative 15𝑥? Well that will be negative five. And let’s just check that what would I need to
multiply negative two by to get positive 10. Well that would also be negative five. So it looks like the factor there, the common factor, is negative five.
So the expression I’ve got now, two 𝑥 times three 𝑥 minus two minus five times
three 𝑥 minus two, if I multiplied out those parenthesis, I would get the line
above and I would get my original expression. But I’ve got an expression involving two 𝑥 times thing, three 𝑥 minus two, and
minus five times thing, three 𝑥 minus two. That same thing is a common factor so I’m gonna that out as a factor.
Now in the first instance, I multiplied three 𝑥 minus two by two 𝑥; and in the
second instance, I multiplied by negative five. So two 𝑥 minus five gives me my second set of parentheses. So the factored form of six 𝑥 squared minus 19𝑥 plus 10 is three 𝑥 minus two times
two 𝑥 minus five.
Well let’s just quickly check it. And three 𝑥 times two 𝑥 is positive six 𝑥 squared; three 𝑥 times negative five is
negative 15𝑥; negative two times positive two 𝑥 is negative four 𝑥; and negative
two times negative five is positive 10.
And just simplifying that down a bit, negative 15𝑥 take away another
fourteen [four] 𝑥 is negative 19𝑥. So the whole thing simplifies to six 𝑥 squared minus 19𝑥 plus 10, which is the
solution that we were looking for, so we’ve got the right answer.
The fully factored form of six 𝑥 squared minus 19𝑥 plus 10 is three 𝑥 minus two
times two 𝑥 minus five. Although of course if you’d have written two 𝑥 minus five times three 𝑥 minus two,
that would definitely have been right as well.