Lehman PTS-3 Summer 2020

1. 𝐏𝐓𝐒 𝟑
Bridge to Calculus Workshop
Summer 2020
Lesson 5
Polynomials
"I have had my results for a long
time: but I do not yet know how I am
to arrive at them.“
- Carl Friedrich Gauss

2. Lehman College, Department of Mathematics
Polynomials (Definitions) (1 of 6)
A polynomial in 𝑥 is an expression of the form:
where 𝑎 𝑛, 𝑎 𝑛−1, 𝑎 𝑛−2, …, 𝑎1, 𝑎0 are real numbers with
𝑎 𝑛 ≠ 0, and 𝑛 is a nonnegative integer (𝑛 ≥ 0).
We call the integer 𝑛 the degree of the polynomial. The
number 𝑎 𝑛 is the leading coefficient, and 𝑎0 is called the
constant term.
Example 1. For the following polynomial, state the
degree, the leading coefficient and the constant term:
Solution.
Constant term:
𝒂 𝒏 𝒙 𝒏 + 𝒂 𝒏−𝟏 𝒙 𝒏−𝟏 + 𝒂 𝒏−𝟐 𝒙 𝒏−𝟐 + ⋯ + 𝒂 𝟏 𝒙 + 𝒂 𝟎
5𝑥3
− 2𝑥2
+ 7𝑥 − 1
3Degree: Leading coefficient: 5
−1

3. Lehman College, Department of Mathematics
Polynomials (Definitions) (2 of 6)
Example 2. Are the following expressions polynomials?
Solution.
Each summand of a polynomial is called a term, e.g.:
The number 3 is called the coefficient of the term, the
number 2 is the degree of the term.
2𝑥4
− 3𝑥−1
+ 2 1
1
7
𝑥3 − 0.0054 𝑥2 + 𝜋 5𝑥2
−
2
𝑥
+ 3
1
3𝑥2 + 2𝑥 + 1
(a) (b)
(c) (d)
(e)
(a) no (b) yes (c) yes (d) no (e) no
𝑥2 − 1
𝑥 − 1
(f)
(f) yes
𝟑𝒙 𝟐

4. Lehman College, Department of Mathematics
Polynomials (Definitions) (3 of 6)
A zero-degree term is called a constant term, e.g.:
A first-degree term is called a linear term, e.g.:
A second-degree term is called a quadratic term, e.g.:
A third-degree term is called a cubic term, e.g.:
A fourth-degree term is called a quartic term, e.g.:
A fifth-degree term is called a quintic term, e.g.:
Polynomials are classified by their degree. For example,
cubic, quadratic, linear, and constant polynomials, etc.
7 = 7𝑥0
3𝑥 = 3𝑥1
5𝑥2
4𝑥3
9𝑥4
2𝑥5

5. Lehman College, Department of Mathematics
Polynomials (Definitions) (4 of 6)
Example 3. Classify the polynomial by degree:
Solution.
A polynomial written with descending powers of the
indeterminate 𝑥 is said to be in standard form.
Example 4. Write the following polynomial in standard
form:
5𝑥4
− 3𝑥2
+ 2 3𝑥 + 2
1
7
𝑥3 −
1
2
𝑥2 + 3 5𝑥2
− 2 + 3
5𝑥5 + 3𝑥 − 9
(a) (b)
(c) (d)
(e)
(a) quartic (b) linear (c) cubic
(d) quadratic (e) quintic
3(f)
(f) constant
5𝑥2 + 3𝑥4 − 9 + 2𝑥

6. Lehman College, Department of Mathematics
Polynomials (Definitions) (5 of 6)
Example 4. Write the following polynomial in standard
form:
Solution.
We also classify polynomials by the number of terms in
the polynomial.
A polynomial with one term is called a monomial, a
polynomial with two terms is called a binomial, and a
polynomial with three terms is called a trinomial.
Polynomials with four or more terms are simply called
polynomials.
Example 5. Classify the following polynomial:
5𝑥2
+ 3𝑥4
− 9 + 2𝑥
3𝑥4
+ 5𝑥2
+ 2𝑥 − 9
5𝑥3 − 2𝑥2

7. Lehman College, Department of Mathematics
Polynomials (Definitions) (6 of 6)
Example 5. Classify the following polynomial:
Answer.
Example 4. Classify the following polynomials:
Solution.
5𝑥3
− 2𝑥2
cubic binomial
5𝑥4
+ 𝑥 − 2 3𝑥 + 2 5𝑥2
− 2𝑥 + 3(a) (b) (c) (d)7𝑥5
(a) quartic trinomial (b) linear binomial
(c) quintic monomial (d) quadratic trinomial

8. Lehman College, Department of Mathematics
Diamonds in the Rough (1 of 1)

9. Lehman College, Department of Mathematics
Operations with Polynomials (1 of 6)
Sums and Differences of Polynomials. We add and
subtract polynomials by grouping like terms.
Example 6. Perform the following polynomial operation:
Solution.
Example 7. Perform the following polynomial operation:
5𝑥3
− 2𝑥2
+ 3𝑥 − 2 + 2𝑥3
+ 3𝑥2
− 5
= 5𝑥3
+ 2𝑥3 + −2𝑥2 + 3𝑥2
+ 3𝑥 + −2 − 5
5𝑥3
− 2𝑥2
+ 3𝑥 − 2 + 2𝑥3
+ 3𝑥2
− 5
= 7𝑥3
+ 𝑥2 + 3𝑥 − 7
5𝑥3
− 2𝑥2
+ 3𝑥 − 2 − 2𝑥3
+ 3𝑥2
− 5
= 5𝑥3 − 2𝑥3 + −2𝑥2
− 3𝑥2
+ 3𝑥 + −2 − −5
= 3𝑥3
− 5𝑥2
+ 3𝑥 + 3

10. Lehman College, Department of Mathematics
Operations with Polynomials (2 of 6)
Product of Polynomials. Before exploring this topic,
we will look at the distributive property of multiplication
over addition.
Example 8. Perform the following algebraic operation:
Solution. Using PEMDAS, we obtain:
However, by the distributive property:
Distributive Property. Let 𝑎, 𝑏, 𝑐 be real numbers, then
Example 9. Show that the following is true (FOIL):
6 ⋅ 4 + 3
6 ⋅ 7 = 42
6 ⋅ 4 + 3 = 6 ⋅ 4 + 6 ⋅ 3 =24 + 18 = 42
𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐 and 𝑎 + 𝑏 𝑐 = 𝑎𝑐 + 𝑏𝑐
𝑎 + 𝑏 𝑐 + 𝑑 = 𝑎𝑐 + 𝑎𝑑 + 𝑏𝑐 + 𝑏𝑑

11. Lehman College, Department of Mathematics
Operations with Polynomials (3 of 6)
Solution. By the distributive property:
Example 9. Prove the difference of two squares rule:
Solution. By the distributive property:
𝑎 + 𝑏 𝑐 + 𝑑 =
= 𝑎𝑐 + 𝑎𝑑 +
𝑎 𝑐 + 𝑑 + 𝑏 𝑐 + 𝑑
𝑏𝑐 + 𝑏𝑑
𝑎 + 𝑏 𝑎 − 𝑏 = 𝑎2
− 𝑏2
𝑎 + 𝑏 𝑎 − 𝑏 = 𝑎 𝑎 − 𝑏 + 𝑏 𝑎 − 𝑏
= 𝑎2
− 𝑎𝑏 + 𝑏𝑎 − 𝑏2
= 𝑎2
− 𝑏2
= 𝑎2 − 𝑎𝑏 + 𝑎𝑏 − 𝑏2

12. Lehman College, Department of Mathematics
Operations with Polynomials (3 of 6)
Example 10. Prove the square of a binomial rule:
Solution. By the distributive property:
𝑎 + 𝑏 𝑎 + 𝑏 = 𝑎 𝑎 + 𝑏 + 𝑏 𝑎 + 𝑏
= 𝑎2 + 𝑎𝑏 + 𝑏𝑎 + 𝑏2
= 𝑎2
+ 2𝑎𝑏 + 𝑏2
𝑎 + 𝑏 2
= 𝑎2
+ 2𝑎𝑏 + 𝑏2

13. Lehman College, Department of Mathematics
Operations with Polynomials (4 of 6)
Example 11. Perform the following operations:
Solution.
Example 12. Use the square of the binomial rule:
to show that:
𝑥 + 2 𝑥 − 3(a) (b) 𝑥 + 4 𝑥 − 4 (c) 𝑥 + 2 2
𝑥 + 2 𝑥 − 3(a) = 𝑥 𝑥 − 3 + 2 (𝑥 − 3)
= 𝑥2
− 3𝑥 + 2𝑥 − 6
= 𝑥2
− 𝑥 − 6
(b) 𝑥 + 4 𝑥 − 4 = 𝑥2
− 42
= 𝑥2
− 16
(c) 𝑥 + 2 2
= 𝑥2
+ 2 𝑥 2 + 22
= 𝑥2
+ 4𝑥 + 4
𝑎 + 𝑏 2
= 𝑎2
+ 2𝑎𝑏 + 𝑏2
𝑎 − 𝑏 2
= 𝑎2
− 2𝑎𝑏 + 𝑏2

14. Lehman College, Department of Mathematics
Operations with Polynomials (5 of 6)
Solution. Given the rule below:
Replace 𝑏 by −𝑏 in both sides of the equation to yield:
Example 12. Use the square of binomial rule to prove
the cube of a binomial rule:
Solution.
𝑎 + 𝑏 2
= 𝑎2
+ 2𝑎𝑏 + 𝑏2
𝑎 − 𝑏 2
=
= 𝑎2 − 2𝑎𝑏 + 𝑏2
𝑎 + (−𝑏) 2
= 𝑎2
+ 2𝑎(−𝑏) + −𝑏 2
𝑎 + 𝑏 3 = 𝑎3 + 3𝑎2 𝑏 + 3𝑎𝑏2 + 𝑏3
𝑎 + 𝑏 3 = (𝑎 + 𝑏) 𝑎 + 𝑏 2 = (𝑎 + 𝑏) 𝑎2
+ 2𝑎𝑏 + 𝑏2
= 𝑎 𝑎2
+ 2𝑎𝑏 + 𝑏2
+ 𝑏 𝑎2
+ 2𝑎𝑏 + 𝑏2
= 𝑎3
+ 2𝑎2
𝑏 + 𝑎𝑏2
+ 𝑎2
𝑏 + 2𝑎𝑏2
+ 𝑏3

15. Lehman College, Department of Mathematics
Operations with Polynomials (6 of 6)
Example 13. Determine the product of the following
binomial and trinomial:
Solution. Use the distributive property:
(2𝑥 + 3)(5𝑥3
− 𝑥 + 4)
2𝑥 + 3 5𝑥3 − 𝑥 + 4 =
= 2𝑥 5𝑥3 − 𝑥 + 4 + 3 (5𝑥3
− 𝑥 + 4)
= 10𝑥4
− 2𝑥2
+ 8𝑥 + 15𝑥3
− 3𝑥 + 12
= 10𝑥4 + 15𝑥3 − 2𝑥2 + 5𝑥 + 12
= 10𝑥4 + 2𝑥(−𝑥) + 8𝑥 + 15𝑥3 + 3(−𝑥) + 12

16. Lehman College, Department of Mathematics
Operations with Polynomials (7 of 7)
Example 14. Determine the product of the following
binomial and trinomial:
Solution. Use the distributive property:
(𝑥 − 3)(𝑥2
− 4𝑥 + 7)
(𝑥2
−4𝑥 + 7)(𝑥 − 3) =
= 𝑥2
𝑥 − 3 − 4𝑥 𝑥 − 3 +
= 𝑥3 − 3𝑥2 − 4𝑥2 + 12𝑥 +
= 𝑥3
− 7𝑥2
+ 19𝑥 − 21
7(𝑥 − 3)
7𝑥 − 21
= 𝑥3
+ 𝑥2
(−3) − 4𝑥2
− 4𝑥(−3) + 7𝑥 + 7(−3)