Find the quotient when six 𝑥 to
the fourth plus 11𝑥 cubed minus 43𝑥 squared minus 40𝑥 is divided by 𝑥 squared
plus 𝑥 minus eight.
So we are finding a quotient, which
means that we’re dividing. And it says we’re taking six 𝑥 to
the fourth plus 11𝑥 cubed minus 43𝑥 squared minus 40𝑥 and dividing it by 𝑥
squared plus 𝑥 minus eight. So to do this, we look at the first
terms. And we think, how many 𝑥 squareds
go into six 𝑥 to the fourth? Or what do we multiply 𝑥 squared
by to make it be six 𝑥 to the fourth?
We would need to multiply 𝑥
squared by six 𝑥 squared. That would give us six 𝑥 to the
fourth. And we put the 𝑥 squared term
above the 𝑥 squared term here. This keeps us organised. Now, we need to begin multiplying
six 𝑥 squared to 𝑥 squared plus 𝑥 minus eight. So six 𝑥 squared times 𝑥 squared
is six 𝑥 to the fourth power.
Now, we take six 𝑥 squared times
𝑥, which is six 𝑥 cubed. And notice we’re putting the 𝑥 to
the fourth powers and 𝑥 cubes underneath each other. And now we take six 𝑥 squared
times negative eight, giving us negative 48𝑥 squared. And we will be subtracting. These cancel. And then, 11𝑥 cubed minus six 𝑥
cubed will be five 𝑥 cubed.
Now, we have negative 43𝑥 squared
minus negative 48𝑥 squared. So really, we’re taking negative
43𝑥 squared plus 48𝑥 squared, which is positive five 𝑥 squared. And we will need to bring down the
So now we’ll repeat this process
again. What do we multiply to 𝑥 squared
to make it be five 𝑥 cubed? We’ll need to multiply 𝑥 squared
by five 𝑥. And now we distribute. So five 𝑥 times 𝑥 squared is five
𝑥 cubed. Five 𝑥 times 𝑥 is five 𝑥
squared. And five 𝑥 times negative eight is
negative 40𝑥. And remember, we’re
So five 𝑥 cubed minus five 𝑥
cubed cancels. Five 𝑥 squared minus five 𝑥
squared cancels. And then negative 40𝑥 minus
negative 40𝑥 is negative 40𝑥 plus 40𝑥. So those cancel as well, which
means there’s nothing left. There’s no remainder. So essentially, this divided
And our quotient, our answer will
be six 𝑥 squared plus five 𝑥.