1. Section 5.3 Scientific Notation Objective: Multiply and divide expressions using scientific notation and exponent properties. Section 5.4 Introduction to Polynomials Objective: Evaluate, add, and subtract polynomials.

2. Section 5.3 Scientific Notation
Scientific notation is used to represent really large or really small numbers.
Scientific Notation has the form:
𝑎×1 0 𝑏 𝑤ℎ𝑒𝑟𝑒 1≤𝑎<10,
𝑏 can be positive or negative integers

3. Steps to convert standard notation to scientific notation
Place the decimal point such that there is one non-zero digit to the left of the decimal point.
Count the number of decimal places that the decimal has "moved" from the original number. This will be the exponent of the 10.
If the original number was less than 1, the exponent is negative; if the original number was greater than 1, the exponent is positive.

4. Examples Standard Notation Scientific Notation 1000256 1.000256× 10 6
The original number was greater than 1, the exponent is positive. The decimal has “moved” 6 places.
9.× 10 −7
The original number was less than 1, the exponent is negative. The decimal has “moved” 7 places.

5. Convert the numbers from scientific notation to standard notation.
To Change from Scientific Notation to Standard Form: (1) Move the decimal point to the right for positive exponents of 10. The exponent tells you how many places to move. (2) Move the decimal point to the left for negative exponents of 10. Again, the exponent tells you how many places to move.

6. Examples Scientific Notation Standard Notation 1.3689× 10 6 1368900
Move the decimal to the right for positive exponents of 10 (6 places). Add zeros if more places are needed.
1.3689× 10 −3
Move the decimal to the left for negative exponents of 10 (3 places). Add zeros if more places are needed.

7. More Examples
Write the number as a whole number or decimal: × 10 −6 =
Write the number in scientific notation:
848,000,000 =8.48× 10 8
Rewrite this number in decimal notation without the use of exponents or scientific notation: × =8,509,000,000,000.

8. Examples Is 3.17200000× 10 42 in scientific notation? If not, why?
Yes.
Is 31.72× in scientific notation? If not, why?
No, because is greater than 9. The decimal is not placed after the first non-zero digit.
Is × in scientific notation? If not, why?
No, because is less than 1. The decimal is not placed after the first non-zero digit.
Is × 10 −2 in scientific notation? If not, why?
Yes.

9. Multiplication and Division in Scientific Notation

10. Examples
Multiply. Give final answer in scientific notation × ⋅ 6.1× 10 −3 = 1.3∙6.1 × −3 =7.93× 10 −2
Divide. Give you answer in scientific notation:
0.8× × = × 10 1−2 =0.2× 10 −1 = 2.× 10 −1 × 10 −1 =2.× 10 −2
The final result in this example is already in scientific notation.
The result 0.2× 10 −1 is not in scientific notation. So we need to work one more step to write the result in scientific notation.

11. Examples
Simplify. Give final answer in scientific notation. (3.0× 10 1 )⋅6.0× 10 −3 9.0× = (3.0∙6.0)× − × = 18.0× 10 −2 9.0× = × 10 −2−5 =2.0× 10 −7

12. Section 5.4 Introduction to Polynomials
Polynomials are made up of terms.
Terms are a product of numbers and variables with nonnegative integer exponents.
Examples:
4 7 𝑥, 5𝑥 𝑦 2 , −3, are polynomials.
The real numbers with the sign are called the coefficients,
such as 4 7 , 5, −3.

13. 4 𝑥 , 5𝑥 𝑦 1 2 , −3 𝑥 are NOT polynomials. Why?
Because: 4 𝑥 =4 𝑥 −1 The exponent is negative. 5𝑥 𝑦 1 2 The exponent of y is not an integer. −3 𝑥 =−3 𝑥 1 2 The exponent of x is not an integer.

14. Introduction to Polynomials
Consider the polynomial: 4 7 𝑥+ 5𝑥 𝑦 2 −3
A term that has only a number (no variables) is called a constant term.
The last term −3 is a constant term.
The real numbers with the sign in each term are called coefficients,
The coefficient of the second term is 5, and the coefficient of the third term is -3.
Terms are connected to each other by addition or subtraction.

15. Special Names for Some Polynomials
Expression are often named based on the number of terms in them.
A monomial has one term, such as 45𝑥, or 3𝑥𝑦, or −9.
A binomial has two terms, such as 45𝑥+1, or 11𝑥−3𝑥𝑦.
A trinomial has three term, such as
45𝑥+𝑥+21, or 11𝑥−3𝑥𝑦−1
The term polynomial means many terms.
Monomials, binomials, trinomials, and expressions with more terms all fall under the umbrella of “polynomials”.

16. Degree of A Polynomial
For a polynomial with one variable, the degree is: The Largest/Highest exponent of that variable. For Example,
Note:
The coefficient of the term that tells the degree of this polynomial is the leading coefficient.
Ex.
4 is the leading coefficient.

17. More Examples Find the degree of the following polynomials:
3𝑥+5 𝑥 6 −7 𝑥 4 has a degree of
−9 𝑦 9 has a degree of
8 has a degree of
5𝑥−8 has a degree of

18. Find the degree of the polynomial
𝑦=5 𝑥 7 −2 𝑥 4 −3 𝑥 6 +6
The degree is:
Select the expressions that are polynomials. There may be more than one correct answer. (The solutions are in green.)
3 𝑥 3 −15 𝑥 2.4 −15 𝑥 6 −13 𝑥 7
6𝑥−18+13 𝑥 28 −6 𝑥 −8 +8 𝑥 29
15 𝑥 𝑥 𝑥 16 −9 𝑥 1 3
𝑥 𝑥 23 −11 𝑥 𝑥 12
2 𝑥 17 −10 𝑥 𝑥 −7.8
−8 𝑥 19

19. Consider the algebraic expression
−3 𝑥 𝑥 4 − 𝑥 −19𝑥+24
How many terms are there? (5 terms)
Identify the constant term. (+24 or 24)
What is the degree? (5)
What is the coefficient of the first term? (-3)
What is the coefficient of the third term? (− 1 10 )

20. Consider the algebraic expression 𝑥 7 −12 𝑥 4 − 𝑥 𝑥 +16 What is the degree of this polynomial? (7) Identify the constant term. (16) Identify the leading coefficient. (1) What is the coefficient of the second term? (-12) What is the coefficient of the third term? (− 1 15 )

21. Evaluate A Polynomial with The Given Value
To evaluate a polynomial, we replace the variable with the given number/value.
Example:
Given the polynomial: −8𝑥− 6𝑥 3 −5 𝑥 Evaluate the polynomial for 𝑥 = 3.
Solution:
−8 3 − −
=−24−6 27 − =−24−162−45+7 =−224

22. Example
Given the polynomial: 5 𝑥 2 +4 𝑥 4 +4 𝑥 3 +3𝑥+5 Evaluate the polynomial for 𝑥=−1 .
5 (−1) 2 +4 (−1) 4 +4 (−1) 3 +3 −1 +5 = −1 −3+5 =5+4−4−3+5 =9−4−3+5 =5−3+5 =2+5 =7

23. Example
Given the polynomial: 7 𝑥 2 −3 𝑦 2 +5𝑥𝑦−5 Evaluate the polynomial for 𝑥=1.7 and 𝑦=−2.6.
7 (1.7) 2 −3 − −2.6 −5 = − −2.6 −5 =20.23−20.08−22.1−5 =−26.95

24. Application Problem
In a northern European country, the formula 𝑦=0.033 𝑥 2 −2.6𝑥 models the number of deaths per year per thousand people, 𝑦, for people who are 𝑥 years old, 40≤𝑥≤60. Approximately how many people per thousand who are 55 years old die each year? For the purposes of this problem, please round your answer to a whole number. Approximately people per thousand who are 55 years old die each year.

25. Solution
It’s actually only asking what y equals to when 𝑥=55 in the formula
𝑦=0.033 𝑥 2 −2.6𝑥+62.55
We substitute 55 for x to get
𝑦=0.033 (55) 2 − 𝑦= y≈19
Approximately people per thousand who are 55 years old die each year.

26. Adding and Subtracting Polynomials
To add or subtract polynomials, we combine the like terms.
What are the like terms?
Terms that have the same variables and each variable has the same exponents.
The coefficients can be the same or different.
Note: Only Like Terms can be combined.

27. For example −2𝑥 𝑦 3 and 5𝑥 𝑦 3 are like terms.
−2 𝑥 3 𝑦 3 and 5𝑥 𝑦 3 are NOT like terms.
Because although the variables are the same, each variable does not have not have the same exponent.
−2𝑥 z 3 and 5𝑥 𝑦 3 are NOT like terms.
Because the variables are not the same.

28. Example
Add the polynomials: 𝑥 5 −6 𝑥 4 −5 𝑥 𝑥 5 +2 𝑥 3 −5 𝑥 = 10 𝑥 5 −6 𝑥 4 −5 𝑥 𝑥 5 +2 𝑥 3 −5 𝑥 2 +6
=19 𝑥 5 −6 𝑥 4 −3 𝑥 3 −5 𝑥 2 +8
The like terms are marked with the same colors. If there are not like terms to combine, you copy them down.

29. Example
Subtract the polynomials and simplify the result completely: −5 𝑥 5 −4 𝑥 3 −6 𝑥 2 +8𝑥 − 10 𝑥 𝑥 4 +2 𝑥 3 −10 =−5 𝑥 5 −4 𝑥 3 −6 𝑥 2 +8𝑥−10 𝑥 5 −11 𝑥 4 −2 𝑥 =−15 𝑥 5 −11 𝑥 4 −6 𝑥 3 −6 𝑥 2 +8𝑥+10
Note that when you subtract a polynomial, you need to change the sign of each term. For example,
𝑥 2 − 2𝑥+𝑥−2 = 𝑥 2 −2𝑥−𝑥+2

30. Perform the indicated operations
10 𝑦 2 −15𝑦−26 − −3 𝑦 2 −11𝑦+12 + −8 𝑦 2 +10𝑦+33 =10 𝑦 2 −15𝑦−26+3 𝑦 2 +11𝑦−12 −8 𝑦 2 +10𝑦+33 = 10 𝑦 2 +3 𝑦 2 −8 𝑦 2 + −15𝑦+11𝑦+10𝑦 +(−26−12+33) =5 𝑦 2 +6𝑦−5

31. Perform the indicated operations −2𝑥+4 − −9 𝑥 2 +4𝑥 𝑥 2 −7 =−2𝑥+4+9 𝑥 2 −4𝑥−6+3 𝑥 2 −7 = 9 𝑥 2 +3 𝑥 2 + −2𝑥−4𝑥 + 4−6−7 =12 𝑥 2 + −6𝑥 + −9 =12 𝑥 2 −6𝑥−9

32. Application Problem
A piece of molding is (113𝑥+96) inches long. If a piece (67𝑥+60) inches is removed, express the length of the remaining piece of molding as a polynomial in 𝑥.
Hint: To be removed, you will use subtraction.
113𝑥+96 − 67𝑥+60
Simply your answer by combing like terms. Your final answer will contain x. That’s ok.