### Video Transcript

Find the remainder when three π₯
cubed minus two π₯ squared plus four π₯ plus five is divided by three π₯ plus
four.

The question is asking us to find
the remainder term when a cubic polynomial is divided by a linear polynomial. One way of doing this is by using
polynomial long division. However, we know this is a long
process. Instead, letβs notice weβre
dividing by a linear polynomial. And weβre only asked to find the
remainder term. This should remind us of the
remainder theorem. We recall the remainder theorems
tells us when π of π₯ is divided by a linear polynomial π₯ minus π, then the
remainder is constant and equal to π evaluated at π.

We do have to be a little careful
with how we use the remainder theorem in this case. We are dividing by a linear
polynomial three π₯ plus four, but itβs not in the form π₯ minus π. So instead of actually doing our
long division, letβs call our quotient polynomial π of π₯ and our remainder
polynomial π of π₯. This means weβll have three π₯
cubed minus two π₯ squared plus four π₯ minus five is equal to three π₯ plus four
times π of π₯ plus π of π₯ for some polynomials π of π₯ and π of π₯. We can see our divisor is a linear
polynomial; it has degree one. Our remainder must have a lower
degree than our divisor. This means it must have degree
zero. In other words, itβs a
constant. Weβll call this constant π.

And at this point, thereβs two
similar ways of solving this equation. If we solve our linear factor is
equal to zero, this gives us π₯ is equal to negative four over three. One way to find the value of π is
to substitute this value directly into this expression. Doing this, we get our cubic
polynomial evaluated at negative four over three is equal to zero times π evaluated
at negative four over three plus π. But this simplifies to just give us
π. And this is a perfectly valid way
of solving this equation. However, weβre going to do this by
taking a factor of three outside of our divisor. Doing this, we can rewrite this as
three times π₯ plus four over three.

And now, weβre starting to see
something interesting. Letβs consider three as part of π
of π₯. So what do we now have? We have our cubic polynomial is
equal to π₯ plus four over three times some polynomial plus a constant. In fact, what weβve done here is
found an expression for our quotient polynomial when we divide our cubic by π₯ plus
four over three. In other words, the remainder term
when we divide by π₯ plus four over three or when we divide by three π₯ plus four
are the same. This means we can just use the
remainder theorem to find our value of π. And weβll do this by evaluating our
cubic polynomial at π₯ is equal to negative four over three. This gives us the following
expression. And then, by evaluating this
expression, we were able to show that our remainder term must be equal to negative
11.