### Video Transcript

Part a) Show that a solution to the
equation 𝑥 cubed minus five 𝑥 plus one equals negative two 𝑥 squared lies between
𝑥 equals zero and 𝑥 equals one.

Part b) Show that the equation 𝑥
cubed minus five 𝑥 plus one equals negative two 𝑥 squared can be rearranged to 𝑥
equals 𝑥 cubed over five plus two 𝑥 squared over five plus one-fifth.

There is also part c), but we’ll
come to that after we’ve done the first two parts. So the first stage is to actually
rearrange a polynomial and make it equal to zero. The reason we’re gonna do that is
so that then we can actually substitute in our values for 𝑥 — so 𝑥 is equal to
zero and 𝑥 is equal to one — and actually see what value the equation will give
us. So if we actually add two 𝑥
squared to both sides of our equation, we’re gonna get 𝑥 cubed plus two 𝑥 squared
minus five 𝑥 plus one is equal to zero.

So now, what we’re gonna do is
actually substitute our value 𝑥 equals zero into our equation. And the reason we’re gonna do this
is because we’re trying to show that the solution to the equation lies between 𝑥
equals zero and 𝑥 equals one. And to do this, we need to actually
see if there’s a change of sign. If we do this and actually
substitute in 𝑥 is equal to zero, we’re gonna get zero cubed plus two multiplied by
zero squared minus five multiplied by zero plus one which will give us a value of
the equation of one. And we get that because we have
zero cubed is zero, so zero plus two multiplied by zero which is zero minus five
multiplied by zero which again is still zero and then plus one which gives us
one.

So now, for the second part, what
we’re gonna do is actually substitute in 𝑥 is equal to one. And when we do this, we get one
cubed plus two multiplied by one squared minus five multiplied by one plus one which
is gonna give us one plus two minus five plus one which gives us a result of
negative one. So then, if we take a look at the
two results we’ve got, we can see that actually when we substitute in 𝑥 equals
zero, we get a positive answer. And if we substitute in 𝑥 equals
one, we get a negative answer. So therefore, between those two
points, there’s been a change of sign. So therefore, that shows that the
solution to the equation must actually lie between these points.

Okay, so that’s part a answered
because as I said we’ve already demonstrated that the solution to the equation 𝑥
cubed minus five 𝑥 plus one equals negative two 𝑥 squared lies between 𝑥 equals
zero and 𝑥 equals one because of the change of sign. Okay, so now, what we need to do is
to actually rearrange our equation to 𝑥 equals 𝑥 cubed over five plus two 𝑥
squared over five plus a fifth.

Now, the key to part b is actually
looking at which term of 𝑥 that’s actually on its own. And we can say it’s the 𝑥 term, so
𝑥 to the power of one as it is, so that’s the term on its own. So therefore, when we’re
rearranging, we want to make sure that that’s left on its own.

So the first stage is to actually
add five 𝑥 to each side. So we get 𝑥 cubed plus one is
equal to negative two 𝑥 squared plus five 𝑥. So then, the next stage is to
actually add two 𝑥 squared to each side of the equation. And we want to do this because as
we said before we want to leave our 𝑥 term on its own. So this is gonna give us 𝑥 cubed
plus two 𝑥 squared plus one equals five 𝑥. So then, what I’ve done is I’ve
actually switched the equation the other way round just as to get it in the same
form that we have in the question. So we got five 𝑥 is equal to 𝑥
cubed plus two 𝑥 squared plus one.

So then, the final stage is to
actually divide each side of the equation by five. So we get 𝑥 is equal to 𝑥 cubed
over five plus two 𝑥 squared over five plus a fifth. So then, we check the question and
this actually matches the exact form that it was looking for. So we’re gonna say that yes, we’ve
actually shown the equation 𝑥 cubed minus five 𝑥 plus one equals negative two 𝑥
squared can be rearranged to 𝑥 equals 𝑥 cubed over five plus two 𝑥 squared over
five plus a fifth.

Okay, let’s move on to part c.

Part c Taking 𝑥 zero to equal
zero, use the iteration formula 𝑥 𝑚 plus one equals 𝑥 𝑛 cubed over five plus two
𝑥 𝑛 squared over five plus one-fifth twice in order to find an estimate for a
solution to the equation 𝑥 cubed minus five 𝑥 plus one equals negative two 𝑥
squared.

So in order to actually understand
what we’re gonna do and how we’d use the iteration formula, let’s think about what
it actually means. Well, what it means is that if we
want to find a term, then it’s gonna be equal to the previous term cubed divided by
five plus two multiplied by the previous term squared divided by five plus
one-fifth. And we’re actually gonna do that
and substitute in our values that we have because we’re told the 𝑥 zero is equal to
zero.

So we’re gonna start off by finding
out the first term, which is 𝑥 one. And we’re gonna do this by
substituting in 𝑥 zero is equal to zero into our iteration formula. So what we’re gonna get is 𝑥 one
is equal to zero cubed over five plus two multiplied by zero squared over five plus
one-fifth. And we did that because as we said,
we substituted in our values for 𝑥 zero which was zero. And this is gonna be equal to a
fifth. And we can actually change that
into a decimal which is 0.2. I’m just gonna use that because
that’s what we’re gonna substitute into our next version of the iteration
formula.

So now, what we’re gonna do is
actually use the fact that we know that 𝑥 one is equal to 0.2 because now we’re
gonna find a second term, so 𝑥 two, and this is like our 𝑥 𝑛 plus one. So now, what we’re gonna do is
actually substitute in our 𝑥 one value of 0.2 into our iteration formula. And when we do that, we get 𝑥 two
is equal to 0.2 all cubed over five plus two multiplied by 0.2 squared over five
plus one-fifth which is gonna give us the fraction 136 over 625. So again, let’s convert this into a
decimal. And when we do, we get 0.2176.

So have we finished here? Well, if we take a look back at the
question, it tells us to actually use the iteration formula twice, which we have
done. So therefore, we can say that
taking 𝑥 zero equals zero and using the iteration formula 𝑥 𝑛 plus one equals 𝑥
𝑛 cubed over five plus two multiplied by 𝑥 𝑛 squared over five plus a fifth,
we’ve actually found an estimate for a solution to equation 𝑥 cubed minus five 𝑥
plus one equals negative two 𝑥 squared of 0.2176.