We have to perform the indicated operation,
because a fraction bar represents division,
we'll perform polynomial long division.
Looking at the numerator or the dividend,
notice how we have x to the fourth
plus 11x cubed
minus 178x plus 15,
because there is no x squared term,
we will include a 0x squared term,
when setting this up for long division,
whenever there are missing terms,
we should always include zero terms,
so the like terms line up vertically.
So for this dividend, we will have x to the fuurth
plus 11x cubed
plus 0x squared
minus 178x
plus 15
and then we have divided by
the divisor of x plus six.
Now to begin the division process,
we only look at the first term of the dividend
and the first term of the divisor
and we ask what times x is equal to x to the fourth,
which is x to the third,
which is the first term of the quotient
and we write the x to the third or x cubed
above the x cubed term of the dividend, which is here.
Another way to determine this term
is to divide x to the fourth
by x, which does give us x to the third
and now we multiply x to the third in the divisor
and write it below the dividend, lining up the like terms,
X cubed times x is x to the fourth
and then x cubed times six is 6x cubed,
giving us plus 6x cubed
and now just like performing long division
with whole numbers, we now subtract,
when performing long division with polynomials,
instead of subtracting, it'll be easier to add the opposite,
which means we can change this to addition,
as long as we change this sign
of the terms inside the parentheses,
which means now we have negative x to the fourth
plus negative 6x to the third
and now adding x to the fourth
plus negative x to the fourth is zero,
11x to the third plus negative 6x to the third
is positive 5x cubed
and now we bring down the next term of the dividend,
which means you bring down plus zero x squared
and now we'll see why we added this zero term,
repeating the process, we only look at the 5x cubed
and the x in the divisor
and ask what times x is equal to 5x cubed,
which is 5x squared,
giving us a plus 5x squared in the quotient.
Okay, if we need to, we can divide 5x cubed
by x, which does give us 5x squared,
now we multiply 5x squared in the divisor,
5x squared times x is 5x cubed
and 5x squared times six is 30x squared,
giving us plus 30x squared.
So notice how I'm multiplying the 5x squared in the divisor,
we do get an x squared term, which again is the reason
why we added the 0x squared term in the dividend,
so that the like terms line up vertically
and now we subtract by adding the opposite,
so we change this to addition,
change this to negative 5x cubed plus negative 30x squared,
adding, here we have opposites and therefore the sum is zero
and then here we have 0x squared plus negative 30x squared,
which gives us negative 30x squared,
bring down the next term,
which means you bring down minus 178x
and now only looking at the negative 30x squared
and the x in the divisor,
we ask what times x is equal to negative 30x squared,
which is negative 30x, giving us a minus 30x in the quotient
or again, if needed, we can divide
negative 30x squared by x,
which does give us negative 30x
and now we multiply negative 30x and the divisor,
negative 30x times x is negative 30x squared
and then negative 30x times six is negative 180x,
giving us minus 180x
and now we subtract by adding the opposite,
so it changes to addition,
changes the positive 30x squared plus 180x
and now adding, here we have opposites again,
this sum is zero, here we have 178x
plus 180x, which gives us positive 2x,
now we bring down the last term in the dividend,
which is the plus 15 and repeat the process one more time,
only looking at the 2x and the x in the divisor,
we ask what times x is equal to 2x,
which is two, giving us a plus two in the quotient
or again, if we need to, we can divide 2x by x,
which does give us two
and now we multiply two in the divisor,
two times x is 2x and we have two times six,
which gives us plus 12
and again, we now subtract by adding the opposites,
so we change this to addition,
we change the sign here and change the sign here,
2x plus negative 2x is zero,
15 plus negative 12 is positive three.
Because we have used all the terms in the dividend,
the positive three here is the remainder,
which we write over the divisor
and therefore the final answer is x to the third
plus 5x squared minus 30x plus two
plus three over the quantity x plus six,
because the remainder is positive three,
we have addition here,
if it was negative, we would have subtraction.
So this is our final answer,
but I do want to mention that sometimes
you may be asked to express the final answer
as the quotient of x cubed plus 5x squared
minus 30x plus two
and then because your remainder is three,
sometimes we'll just say the remainder R is equal to three.
So just make sure you give the final answer
in the form requested.
I hope you found this helpful.