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
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
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 positive10.
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.