1) The document provides examples and explanations of polynomial long division. It demonstrates how to perform long division with polynomial expressions by distributing terms and subtracting multiples of the denominator polynomial from the numerator polynomial. 2) It also discusses tricks for polynomial long division, such as creating multiples of the denominator polynomial to directly obtain terms of the quotient. 3) The document features biographies of mathematicians Paul Erdos and Terence Tao, including a photo of them working together when Tao was 10 years old.

1. 𝐏𝐓𝐒 𝟑
Bridge to Calculus Workshop
Summer 2020
Lesson 22
Polynomial Long
Division
"Perfect numbers like perfect men
are very rare.“ – Descartes -

2. Lehman College, Department of Mathematics
Factoring Binomials (1 of 1)
Example 1. Factor the following expression completely:
Solution. Note that the greatest common integer factor
is 6 and the highest common power of 𝑥 is:
24𝑥4
− 54𝑥8
𝑥4
24𝑥4
− 54𝑥8
= 6𝑥4
4 − 9𝑥4
= 6𝑥4
22
− 3𝑥2 2
= 6𝑥4
2 − 3𝑥2
2 + 3𝑥2
= 6𝑥4
2 − 𝑥 3 2 + 𝑥 3 2 + 3𝑥2

3. Lehman College, Department of Mathematics
Factoring Binomials (1 of 1)
Example 2. Factor the following expression completely:
Solution. Note that the greatest common integer factor
is 1 and the highest common power of 𝑥 is:
Here, 𝑥2
+ 1 is an irreducible quadratic.
Is 𝑥4 + 1 irreducible?
No, but it is the product of two irreducible quadratics.
𝑥9
− 𝑥
𝑥
𝑥9
− 𝑥 = 𝑥 𝑥8 − 1
= 𝑥 𝑥4
− 1 𝑥4
+ 1
= 𝑥 𝑥2
− 1 𝑥2
+ 1 𝑥4
+ 1
= 𝑥 𝑥 − 1 𝑥 + 1 𝑥2 + 1 𝑥4 + 1

4. Lehman College, Department of Mathematics
Rational Numbers (1 of 2)
Let us look at integer multiplication. For example, if:
then we say 2 and 3 are integer factors of 6. Now, we
introduce a new operation called integer division. If we
have the product 2 ⋅ 3 = 6, then we define the quotients:
Let us extend the concept to quotients, such as:
Since there are no integers 𝑚 and 𝑛, such that:
Then the above quotients are not integers.
2 ⋅ 3 = 6
6
3
= 2
6
2
= 3and
5
3
6
4
and
3 ⋅ 𝑛 = 5 4 ⋅ 𝑚 = 6and

5. Lehman College, Department of Mathematics
Rational Numbers (2 of 2)
Let 𝑝 and 𝑞 be integers with 𝑞 ≠ 0, then the quotient:
Is called a rational number. The adjective rational
comes from the word ratio, meaning quotient.
Why is division by zero not allowed? Suppose we have:
Where 𝑟 is a rational number. Then 𝑟 ⋅ 0 = 2, but the
product of any number with zero is zero, so no such
number 𝑟 can exist, and the quotient is thus undefined.
How about the quotient
0
0
? Suppose
0
0
= 𝑟, so 𝑟 ⋅ 0 = 0.
In this case, 𝑟 could be any real, and is thus undefined.
𝑝
𝑞
2
0
= 𝑟

6. Lehman College, Department of Mathematics
Integer Division (1 of 6)
Example 1. Perform the following operation:
Solution. Since:
Then:
The number 1 is called the quotient, and 2 is the
remainder of integer division.
5
3
3 ⋅ 1 = 3 3 ⋅ 2 = 6and< 5 > 5
5
3
=
3 + 2
3
=
3
3
+
2
3
= 1 +
2
3

7. Lehman College, Department of Mathematics
Rational Function Definition (1 of 1)
Let 𝑝(𝑥) and 𝑞(𝑥) be polynomials with 𝑞(𝑥) ≠ 0, then
the quotient:
is called a rational function. For example, the following
are all rational functions:
𝑝(𝑥)
𝑞(𝑥)
1
𝑥
𝑥 + 1
𝑥2 + 3𝑥 − 4
5𝑥3
+ 2𝑥 + 3
3𝑥2 + 2𝑥 − 1
(a) (b) (c)
5(d)
𝑥 + 1
𝑥 − 1
(e) 4𝑥2 + 2𝑥 + 5(f)

8. Lehman College, Department of Mathematics
Polynomial Long Division (1 of 8)
Example 2. Perform the following operation:
Solution. We distribute the product:
Similar to integer division, it follows that:
(3𝑥 + 5)(𝑥 + 2)
3𝑥 + 5 𝑥 + 2 = 3𝑥 𝑥 + 2 + 5 𝑥 + 2
= 3𝑥2 + 6𝑥 + 5𝑥 + 10
= 3𝑥2
+ 11𝑥 + 10
3𝑥2
+ 11𝑥 + 10
𝑥 + 2
= 3𝑥 + 5
3𝑥2
+ 11𝑥 + 10
3𝑥 + 5
= 𝑥 + 2

9. Lehman College, Department of Mathematics
Polynomial Long Division (2 of 8)
Example 3: In a rational function, if the numerator
polynomial is the same or of higher degree than the
denominator, we can perform polynomial long-division:
It follows that:
3𝑥2
+ 11𝑥 + 10𝑥 + 2
−(3𝑥2 + 6𝑥)
5𝑥
3𝑥
− (5𝑥 + 10)
0
+ 5
+ 10
3𝑥2 + 11𝑥 + 10
𝑥 + 2
= 3𝑥 + 5
Quotient
Remainder

10. Lehman College, Department of Mathematics
Polynomial Long Division (3 of 8)
Example 4. Perform the following operation:
Solution. We distribute the product:
Similar to integer division, it follows that:
(𝑥 − 1)(𝑥 + 1)
(𝑥 − 1)(𝑥 + 1) = 𝑥 𝑥 + 1 − 1 𝑥 + 1
= 𝑥2 + 𝑥 − 𝑥 − 1
= 𝑥2
− 1
𝑥2 − 1
𝑥 − 1
= 𝑥 + 1
𝑥2 − 1
𝑥 + 1
= 𝑥 − 1and

11. Lehman College, Department of Mathematics
Polynomial Long Division (4 of 8)
Example 5: Perform polynomial long division:
Solution. Since the numerator polynomial is of higher
degree than the denominator, we will perform
polynomial long-division:
𝑥2
− 1
𝑥 + 1
𝑥2
+ 0𝑥 − 1𝑥 + 1
−(𝑥2
+ 𝑥)
𝑥
−(− 𝑥 − 1)
0
− 1
− 1−𝑥
Quotient
Remainder

12. Lehman College, Department of Mathematics
Erdős and Tao (1 of 2)
Terence Tao (b. 1975)
- Australian-American Mathematician
Paul Erdős (1913-1996)
- Hungarian Mathematician

13. Lehman College, Department of Mathematics
Erdős and Tao (2 of 2)
A 10-year-old Terence Tao hard at work with Paul
Erdős in 1985. Courtesy of Wikimedia Commons.

14. Lehman College, Department of Mathematics
Polynomial Long Division (5 of 8)
Example 6: Perform polynomial long division:
Solution. Since the numerator polynomial is of higher
degree than the denominator, we will perform
polynomial long-division:
𝑥3
− 𝑥 + 3
𝑥2 + 𝑥 − 2
𝑥3 + 0𝑥2 − 𝑥 + 3𝑥2
+ 𝑥 − 2
−(𝑥3
+ 𝑥2
− 2𝑥)
−𝑥2 + 𝑥
𝑥
−(−𝑥2 − 𝑥 + 2)
2𝑥 + 1
− 1
+ 3
Quotient
Remainder

15. Lehman College, Department of Mathematics
Polynomial Long Division (6 of 8)
Solution (cont’d). It follows that:
𝑥3
− 𝑥 + 3
𝑥2 + 𝑥 − 2
= 𝑥 − 1 +
2𝑥 + 1
𝑥2 + 𝑥 − 2

16. Lehman College, Department of Mathematics
Polynomial Long Division (7 of 8)
Example 6: Perform polynomial long division:
Solution. :
2𝑥4
+ 3𝑥3
+ 5𝑥 − 1
𝑥2 + 3𝑥 + 2
2𝑥4
+ 3𝑥3
+ 0𝑥2
+ 5𝑥 − 1𝑥2
+ 3𝑥 + 2
−(2𝑥4
+ 6𝑥3
+ 4𝑥2
)
−3𝑥3
− 4𝑥2
2𝑥2
−(−3𝑥3
− 9𝑥2
− 6𝑥)
5𝑥2 + 11𝑥
− 3𝑥
+ 5𝑥
Quotient
Remainder
− 1
+ 5
− 5𝑥2
+ 15𝑥 + 10
−4𝑥 − 11

17. Lehman College, Department of Mathematics
Polynomial Long Division (8 of 8)
Solution (cont’d). It follows that:
2𝑥4
+ 3𝑥3
+ 5𝑥 − 1
𝑥2 + 3𝑥 + 2
= 2𝑥2
− 3𝑥 + 5 +
−4𝑥 − 11
𝑥2 + 3𝑥 + 2

18. Lehman College, Department of Mathematics
Polynomial Long Division Tricks (1 of 3)
Example 7. Perform polynomial long division:
Solution: Create multiples of the denominator:
𝑥2
𝑥2 + 1
𝑥2
𝑥2 + 1
=
𝑥2
+ 1 − 1
𝑥2 + 1
=
𝑥2
+ 1
𝑥2 + 1
−
1
𝑥2 + 1
= 1 +
1
𝑥2 + 1

19. Lehman College, Department of Mathematics
Polynomial Long Division Tricks (2 of 3)
Example 8. Perform polynomial long division:
Solution. Create multiples of the denominator:
𝑥2
+ 2𝑥 + 3
𝑥 + 1
𝑥2 + 2𝑥 + 3
𝑥 + 1
=
𝑥2
+ 𝑥 + 𝑥 + 3
𝑥 + 1
=
𝑥2 + 𝑥
𝑥 + 1
+
𝑥 + 3
𝑥 + 1
= 𝑥 +
𝑥 + 1 + 2
𝑥 + 1
= 𝑥 + 1 +
2
𝑥 + 1