We have seen simple division of polynomials in Algebra 1, such as:

Division with factoring the numerator and reducing:divide1
Division with creating separate fractions and reducing:
divide2
Dividing a factorable polynomial by a binomial:
divide3

It is now time to expand our division skills to allow us to conquer more situations.

bullet Long Division



Our first new strategy is LONG DIVISION. Yes, it is the same idea that you learned in elementary school. Let's see how our new process resembles what you already know how to do.

Numerical Long Division Process:
divide5c
Algebraic Long Division Process:

divide6
There is no remainder to
be listed in this problem.
Division Algorithm:
divideruleELEM
More formally stated, the division algorithm algebraically states that:

if f (x) and d (x)≠0 are polynomials such that the degree of d (x) ≤ the degree of f (x),
then there exists unique polynomials q (x) and r (x) such that
divide13
and the degree of r (x) < the degree of d (x).
If r (x) = 0, then d (x) divides evenly into f (x), making d (x) a factor of f (x).

This algorithm is simply saying that when the two polynomials are divided (f (x) ÷ d (x)), the solution will be the quotient, q(x), plus a remainder expressed as the remainder over the divisor, r(x)/d(x).

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Let's examine algebraic long division in a variety of situations. We will be assuming that the divisors in these examples are not zero (i.e., in Example 1, assume x - 3 ≠0).

pin1
Divide: (2x3 + 4x2 + 5x - 1) by (x - 3)
Process:
1.
Start the division by asking, "What term multiplied times x will give 2x3 ? "
In this problem, the answer is 2x2.
divide8
2. Multiply your answer to this question times the divisor (x - 3), lining up the similar terms. Then subtract (being careful to change the signs).
3. Bring down the next available term in the dividend.
4. Repeat this process (from step 1 through step 3) until all terms from the dividend have been used.
5. If there is a remainder, place the remainder over the divisor and add it to your quotient answer. (This is the same manner of expressing the remainder that you saw in elementary long division.)

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pin2 beware
Divide: (3x2 + x3 - 2x + 6) by (x - 1)
Be sure that the polynomial is in descending order (by powers).
(x3 + 3x2 - 2x + 6) by (x - 1)
If not, re-write the polynomial into descending order before beginning the division.
divide7
division dude

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pin3 beware
Divide: (2x4 + 4x2 - 1) by (x + 1)
Look for any missing terms in the descending order.
2x4 + 0x3 + 4x2 + 0x - 1
Replace missing terms with a coefficient of 0 to "hold" their location in the division process.
divide9
divisiondude2

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pin4
Divide: divide16
Remember to replace missing terms with 0 coefficients to "hold" their positions in the division.
divide17
Did you notice that this division problem is an example of factoring the difference of perfect cubes?
Remember the formula:

a
3 - b3 = (a - b)(a2 + ab + b2)

In this problem, a3 = a3 and b3 = 8 (making b = 2).
a3 - 8 = (a - 2)(a2 + 2a + 4)


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pin5 beware
(x6 + 2x5 +5x3 +4x 2 + 6) ÷ (x3 + 2)
Be sure to line-up the similar terms before subtracting when the divisor contains a power greater than 1.
divide10
When preparing to subtract, be sure that your terms are lined-up under similar terms.
Notice in this problem that we replaced the missing terms with zero coefficients to "hold" their places in the division, but in effect, these missing terms were never utilized in this problem.

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pin6
divide11b
Multiple variables:
divide12
Follow the same process when working with multiple variables.
Since a2 + 2ab + b2 = (a + b)2, the division in this problem has shown that:
divide12bb
We will be seeing this "pattern" for the expansion of (a + b)3 again.

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pin7
Problem with Different wording:
QUESTION:divide15d
SOLUTION:divide15t
This question is asking you to divide the two given polynomials and to express the answer as a quotient plus a remainder divided by the divisor. In other words, it is simply asking you to divide these polys as we have been doing.
formulapic

 

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