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AP CALCULUS
Review 5: Long Division and Completing the
Square
Polynomial Long Division
1. Divide
467 12
Steps
1) Divide
2) Multiply
3) Subtract
4) Bring Down
5) Repeat
2.
x 3  7 x 2  7 x  6  x  2
Step1:Divide.We will divide the first term
of the polynomial by x
2.
x 3  7 x 2  7 x  6  x  2
Step 2: Multiply. We multiply the piece we
just put as part of the answer x 2  by the
entire binomial x  2 . This is written
underneath the original polynomial (just
like we would in an arithmetic long division
problem.
2.
x 3  7 x 2  7 x  6  x  2
Step 3: Subtract. We now need to
subtract x 3  2 x  .Remember that to
subtract a polynomial you have to
change the sign of each term then
combine like terms as shown here:
2.
x 3  7 x 2  7 x  6  x  2
Step 4: Bring down. Simply bring down the next
term in the polynomial:
2.
x 3  7 x 2  7 x  6  x  2
Step 5: Repeat. Now we start back again at the
beginning.
Divide: Now we will divide the first term of our
answer 5x2 by x., so 5x goes at the top as part of
our answer:
2.
x 3  7 x 2  7 x  6  x  2
Multiply: Now we multiply the piece we just
put as part of the answer (5x) by the entire binomial
(x+2). This is written underneath (just like we
would in an arithmetic long division problem).
2.
x 3  7 x 2  7 x  6  x  2
Subtract: We now need to subtract 5 x 2  10 x
Remember that to subtract a polynomial you have
to change the sign of each term then combine like
terms as shown here:
2.
x 3  7 x 2  7 x  6  x  2
5 x 2  10 x
Bring Down: Simply bring down the next term
in the polynomial:
2.
x 3  7 x 2  7 x  6  x  2
Repeat: Once again we start back at the beginning
with division
Divide: Now we will divide the first term of our
answer -3x by x.  3 x
x
 3
so -3 goes at the top as part of our answer:
2.
x 3  7 x 2  7 x  6  x  2
Multiply: Now we multiply the piece we just put as
part of the answer (-3) by the entire binomial (x+2).3(x + 2) = -3x -6. This is written underneath the
original polynomial (just like we would in an
arithmetic long division problem.
2.
x 3  7 x 2  7 x  6  x  2
Step 6: Subtract. We now need to subtract -3x-6.
Remember that to subtract a polynomial you have to
change the sign of each term then combine like terms
as shown here:
Note: no remainder so we
have our answer x 2  5 x  3
Divide using Polynomial Long Division


3. 2 x  8 x  11x  5  x  3
3
2
Completing the Square
The process of making a
quadratic expression
into a perfect square by
adding half the
coefficient of the x term
squared is called
completing the square.
Completing the square
The process by which a quadratic can factor
4.
2

5
x  10 x
2
x  5
5.
6.
2
x 2  8x  4 2
2
x  4 
2
x 2  20 x  10
 x  10 
2
Complete the square to find the vertex form.
7.
f ( x)  x  10 x  24
2
Graph, identify vertex
Complete the square to find the vertex form
2
8. f ( x)  2 x  8x  7
Complete the square to find the inverse trig form
8.
1
f ( x)  2
x  4 x  10
Assignment
Review Assignment 5