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# Dividing polynomials with remainders: monomial divisors (example 2)

Video transcript

The quotient of two
polynomials-- a of x and b of x-- can be
written in the form a of x over b of x is equal to
q of x plus r of x over b of x-- where q of x and r
of x are polynomials and the degree of r of x is
less than the degree of b of x. Write the quotient
7x to the sixth plus x to the third plus 2x plus
1 over x squared in this form. Well, this one is
pretty straightforward because we're
dividing by x squared. So you could literally
view this as 7x to the sixth
divided by x squared plus x to the third divided
by x squared plus 2x divided by x squared plus 1
divided by x squared. So we could just do
this term by term. What's 7x to the sixth
divided by x squared? Well, x to the sixth divided by
x squared is x to the fourth. So it's going to be 7x
to the fourth power. And then, same thing
right over here. Plus x to the third
divided by x squared. Well, that's just going to be x. So plus x. And then, we're going to
have 2x divided by x squared. But remember, we
want to write it in a form of r of x over
b of x-- where r of x has a lower degree than b of x. Well, 2x has a lower
degree than x squared. Here this is degree 1. This is degree 2. So you could write it as
plus 2x over x squared. Like that. And then, you could write
plus 1 over x squared. So you could do this--
plus 1 over x squared. So you could write it like that. But that's not exactly
the form that they want. They want us to write it q
of x-- and you could view that as 7x to the fourth plus x. And then, they want plus
r of x over b of x So plus some polynomial over
x squared in this case. So instead of writing it as 2x
over x squared plus 1 over x squared, we could just write
it as 2x plus 1 over x squared. So one way to think about it. So let me just put
some parentheses here so that it interprets
my typing correctly. So notice, this part
of the polynomial, these terms have an equal or
higher degree than x squared. So I just divided those. 7x to the sixth divided by x
squared is 7x to the fourth. x to the third divided
by x squared is x. And then, once I got two terms
that had a lower degree than x squared, I just left on there. I just said plus whatever 2x
plus 1 divided by x squared is. And that's the form that they
wanted us to write it in. We'll check our answer. And we got it right.