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Long Division of Polynomials
Long Division of Polynomials
Terri Miller
February 16, 2009
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Problem
Divide the polynomial
P(x) = 3x 4 + 2x 3 − x + 2
by the polynomial
D(x) = x 2 + 2x − 1
x 2 + 2x − 1
3x 4 + 2x 3
−x +2
Note the space left for the x 2 term, you could also put 0x 2 to hold
this spot. Be sure to leave a spot for every power of x from the
degree of the polynomial down to the constant.
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Begin by comparing the leading term of the divisor, D(x), to the
leading term of the dividend, P(x).
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Begin by comparing the leading term of the divisor, D(x), to the
leading term of the dividend, P(x).
Note that x 2 goes into 3x 4 , 3x 2 times; i.e. x 2 · 3x 2 = 3x 4 . So our
first step is to enter that as the beginning of the quotient.
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Begin by comparing the leading term of the divisor, D(x), to the
leading term of the dividend, P(x).
Note that x 2 goes into 3x 4 , 3x 2 times; i.e. x 2 · 3x 2 = 3x 4 . So our
first step is to enter that as the beginning of the quotient.
3x 2
x 2 + 2x − 1
3x 4 + 2x 3
Terri Miller
−x +2
Long Division of Polynomials
Long Division of Polynomials
Steps
Next multiply the divisor by 3x 2 and put the result under the
dividend.
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Next multiply the divisor by 3x 2 and put the result under the
dividend.
3x 2
x 2 + 2x − 1
3x 4 + 2x 3
− 3x 4 − 6x 3 + 3x 2
Terri Miller
−x +2
Long Division of Polynomials
Long Division of Polynomials
Steps
Next multiply the divisor by 3x 2 and put the result under the
dividend.
3x 2
x 2 + 2x − 1
3x 4 + 2x 3
− 3x 4 − 6x 3 + 3x 2
−x +2
Then, just as for numbers, we subtract this result from the
dividend and bring down the next term.
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Next multiply the divisor by 3x 2 and put the result under the
dividend.
3x 2
x 2 + 2x − 1
3x 4 + 2x 3
− 3x 4 − 6x 3 + 3x 2
−x +2
Then, just as for numbers, we subtract this result from the
dividend and bring down the next term.
3x 2
x 2 + 2x − 1
3x 4 + 2x 3
− 3x 4 − 6x 3 + 3x 2
− 4x 3 + 3x 2
Terri Miller
−x +2
−x
Long Division of Polynomials
Long Division of Polynomials
Steps
Now compare the leading term of the divisor, x 2 , to the leading
term of this result, −4x 3 .
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Now compare the leading term of the divisor, x 2 , to the leading
term of this result, −4x 3 .
Note that x 2 goes into −4x 3 , −4x times; i.e. x 2 · −4x = −4x 3 .
Enter that as the next term of the quotient.
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Now compare the leading term of the divisor, x 2 , to the leading
term of this result, −4x 3 .
Note that x 2 goes into −4x 3 , −4x times; i.e. x 2 · −4x = −4x 3 .
Enter that as the next term of the quotient.
3x 2 − 4x
x 2 + 2x − 1
3x 4 + 2x 3
− 3x 4 − 6x 3 + 3x 2
− 4x 3 + 3x 2
Terri Miller
−x +2
−x
Long Division of Polynomials
Long Division of Polynomials
Steps
Multiply the divisor by −4x and put the result under the result of
the subtraction.
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Multiply the divisor by −4x and put the result under the result of
the subtraction.
3x 2 − 4x
x 2 + 2x − 1
3x 4 + 2x 3
− 3x 4 − 6x 3 + 3x 2
−x +2
− 4x 3 + 3x 2 − x
4x 3 + 8x 2 − 4x
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Again, we subtract this result from the line above it and bring
down the next term.
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Again, we subtract this result from the line above it and bring
down the next term.
3x 2 − 4x
x 2 + 2x − 1
3x 4 + 2x 3
− 3x 4 − 6x 3 + 3x 2
−x +2
− 4x 3 + 3x 2 − x
4x 3 + 8x 2 − 4x
11x 2 − 5x + 2
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Now we note that x 2 goes into 11x 2 , 11 times. Enter that as the
next term of the quotient.
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Now we note that x 2 goes into 11x 2 , 11 times. Enter that as the
next term of the quotient.
3x 2 − 4x + 11
x 2 + 2x − 1
3x 4 + 2x 3
− 3x 4 − 6x 3 + 3x 2
−x +2
− 4x 3 + 3x 2 − x
4x 3 + 8x 2 − 4x
11x 2 − 5x + 2
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Multiply the divisor by 11 and put the result under the previous
line.
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Multiply the divisor by 11 and put the result under the previous
line.
3x 2 − 4x + 11
x 2 + 2x − 1
3x 4 + 2x 3
−x +2
4
3
2
− 3x − 6x + 3x
− 4x 3 + 3x 2 − x
4x 3 + 8x 2 − 4x
11x 2 − 5x + 2
− 11x 2 − 22x + 11
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Again, subtract this result from the line above it and bring down
the next term.
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Steps
Again, subtract this result from the line above it and bring down
the next term.
3x 2 − 4x + 11
x 2 + 2x − 1
3x 4 + 2x 3
− 3x 4 − 6x 3 + 3x 2
−x +2
− 4x 3 + 3x 2 − x
4x 3 + 8x 2 − 4x
11x 2 − 5x + 2
− 11x 2 − 22x + 11
− 27x + 13
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Solution
Since x 2 , the leading term of the divisor, is of higher degree than
the result of our last step, we are done.
Solution
3x 2 − 4x + 11
x 2 + 2x − 1
3x 4 + 2x 3
− 3x 4 − 6x 3 + 3x 2
−x +2
− 4x 3 + 3x 2 − x
4x 3 + 8x 2 − 4x
11x 2 − 5x + 2
− 11x 2 − 22x + 11
− 27x + 13
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Identify the parts
As with long division of numbers, the parts of the problem have
terms to identify them.
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Identify the parts
As with long division of numbers, the parts of the problem have
terms to identify them.
D(x) = x 2 + 2x − 1 is called the divisor
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Identify the parts
As with long division of numbers, the parts of the problem have
terms to identify them.
D(x) = x 2 + 2x − 1 is called the divisor
P(x) = 3x 4 + 2x 3 − x + 2 is called the dividend
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Identify the parts
As with long division of numbers, the parts of the problem have
terms to identify them.
D(x) = x 2 + 2x − 1 is called the divisor
P(x) = 3x 4 + 2x 3 − x + 2 is called the dividend
Q(x) = 3x 2 − 4x + 11 is called the quotient
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Identify the parts
As with long division of numbers, the parts of the problem have
terms to identify them.
D(x) = x 2 + 2x − 1 is called the divisor
P(x) = 3x 4 + 2x 3 − x + 2 is called the dividend
Q(x) = 3x 2 − 4x + 11 is called the quotient
R(x) = −27x + 13 is called the remainder
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Rewrite the polynomial P(x)
As a result of this division, we can rewrite the polynomial:
P(x) = Q(x) · D(x) + R(x)
or, equivalently
P(x)
R(x)
= Q(x) +
.
D(x)
D(x)
Terri Miller
Long Division of Polynomials
Long Division of Polynomials
Rewrite the polynomial P(x)
As a result of this division, we can rewrite the polynomial:
P(x) = Q(x) · D(x) + R(x)
or, equivalently
P(x)
R(x)
= Q(x) +
.
D(x)
D(x)
For our particular division problem this looks like:
3x 4 + 2x 3 − x + 2 = (3x 2 − 4x + 11)(x 2 + 2x − 1) + (−27x + 13)
or
−27x + 13
3x 4 + 2x 3 − x + 2
= 3x 2 − 4x + 11 + 2
x 2 + 2x − 1
x + 2x − 1
Terri Miller
Long Division of Polynomials