Matho083 Bianco Warm Up Multiply: 1) (x2) (x3) 2) (3x2) (4x3) 3) (4x5) (-2x3) 4) (5xy2) (2x3y)

Scientific Notation In many fields of science we encounter very large or very small numbers. Scientific notation is a convenient shorthand for expressing these types of numbers. A positive number is written in scientific notation if it is written as a product of a number a, where 1  a < 10, and an integer power r of 10. a  10r Scientific notation

Scientific Notation Writing a Number in Scientific Notation Move the decimal point in the original number to the until the new number has a value between 1 and 10. Count the number of decimal places the decimal point was moved in Step 1. If the original number is 10 or greater, the count is positive. If the original number is less than 1, the count is negative. Write the product of the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2.

Scientific Notation Example: 4700 4700 = 4.7  103 0.00047 Write each of the following in scientific notation. Have to move the decimal 3 places to the left, so that the new number has a value between 1 and 10. 4700 1) Since we moved the decimal 3 places, and the original number was > 10, our count is positive 3. 4700 = 4.7  103 Have to move the decimal 4 places to the right, so that the new number has a value between 1 and 10. 0.00047 2) Since we moved the decimal 4 places, and the original number was < 1, our count is negative 4. 0.00047 = 4.7  10-4

Scientific Notation Writing a Scientific Notation Number in Standard Form Move the decimal point the same number of places as the exponent on 10. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.

Scientific Notation Example: 5.2738  103 5.2738  103 = 5273.8 Write each of the following in standard notation. 5.2738  103 1) Since the exponent is a positive 3, we move the decimal 3 places to the right. 5.2738  103 = 5273.8 6.45  10-5 2) Since the exponent is a negative 5, we move the decimal 5 places to the left. 00006.45  10-5 = 0.0000645

SCIENTIFIC TO STANDARD Exp positive = move decimal right Exp negative = move decimal left

STANDARD TO SCIENTIFIC Move decimal right = negative exp Move decimal left = positive exp

Simplify: (7x5)(x3) [(32)4]3 (p2q8)(p2q) (3pq6)2 JOURNAL Simplify: (7x5)(x3) [(32)4]3 (p2q8)(p2q) (3pq6)2

PRODUCT OF SCIENTIFIC NOTATION Multiply the numbers Add the exponents

EXAMPLE (4.11 x 1013) x (3.78 x 10-5)

EXAMPLE #2 (9 x 104) x (4 x 106)

QUOTIENT OF SCIENTIFIC NOTATION Divide the numbers Subtract the exponents

EXAMPLE (4.11 x 1013) (3.78 x 10-5)

EXAMPLE #2 (9 x 104) (4 x 106)

1. Evaluate if x = 3 and y = -5 x3 + y4 2. Multiply x5y2 - x2y5 3. Mult 3x3 (-2 x5 )

Polynomials and Polynomial Functions § 5.3 Polynomials and Polynomial Functions

Polynomial Vocabulary Term – a number or a product of a number and variables raised to powers Coefficient – numerical factor of a term Constant – term which is only a number Polynomial is a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator.

Polynomial Vocabulary In the polynomial 7x5 + x2y2 – 4xy + 7 There are 4 terms: 7x5, x2y2, -4xy and 7. The coefficient of term 7x5 is 7, of term x2y2 is 1, of term –4xy is –4 and of term 7 is 7. 7 is a constant term.

Types of Polynomials Monomial is a polynomial with one term. Binomial is a polynomial with two terms. Trinomial is a polynomial with three terms.

Degrees Degree of a term Degree of a polynomial To find the degree, take the sum of the exponents on the variables contained in the term. Degree of a constant is 0. Degree of the term 5a4b3c is 8 (remember that c can be written as c1). Degree of a polynomial To find the degree, take the largest degree of any term of the polynomial. Degree of 9x3 – 4x2 + 7 is 3.

Combining Like Terms Example: Like terms are terms that contain exactly the same variables raised to exactly the same powers. Warning! Only like terms can be combined through addition and subtraction. Example: Combine like terms to simplify. x2y + xy – y + 10x2y – 2y + xy = x2y + 10x2y + xy + xy – y – 2y (Like terms are grouped together) = (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y = 11x2y + 2xy – 3y

Adding Polynomials Adding Polynomials To add polynomials, combine all the like terms. Example: Add. (3x – 8) + (4x2 – 3x +3) = 3x – 8 + 4x2 – 3x + 3 = 4x2 + 3x – 3x – 8 + 3 = 4x2 – 5

Subtracting Polynomials To subtract polynomials, add its opposite. Example: Subtract. 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4 = y + 8 (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7) = – a2 + 1 – a2 + 3 + 5a2 – 6a + 7 = – a2 – a2 + 5a2 – 6a + 1 + 3 + 7 = 3a2 – 6a + 11

Adding and Subtracting Polynomials In the previous examples, after discarding the parentheses, we would rearrange the terms so that like terms were next to each other in the expression. You can also use a vertical format in arranging your problem, so that like terms are aligned with each other vertically.

Types of Polynomials a > 0 a < 0 Using the degree of a polynomial, we can determine what the general shape of the function will be, before we ever graph the function. A polynomial function of degree 1 is a linear function. We have examined the graphs of linear functions in great detail previously in this course and prior courses. A polynomial function of degree 2 is a quadratic function. In general, for the quadratic equation of the form y = ax2 + bx + c, the graph is a parabola opening up when a > 0, and opening down when a < 0. a > 0 x a < 0 x

Types of Polynomials Polynomial functions of degree 3 are cubic functions. Cubic functions have four different forms, depending on the coefficient of the x3 term. x3 coefficient is negative x3 coefficient is positive

POLYNOMIALS MULTIPLY: 1. (7) (2x - 5) (7) (2x) - (7) (5) 14x - 35

2. (2x2)(5x4 + 7x) (2x2) (5x4)+ (2x2) (7x) 10x6 + 14x3 POLYNOMIALS MULTIPLY: 2. (2x2)(5x4 + 7x) (2x2) (5x4)+ (2x2) (7x) 10x6 + 14x3

Multiplying by FOIL (3x + 2) (5x + 4) F O I L

F O I L (3x + 2) (5x + 4) 15x2 Multiplying by FOIL F ---> First Terms

F O I L (3x + 2) (5x + 4) 15x2 + 12x Multiplying by FOIL O ---> Outer Terms 15x2 + 12x

F O I L (3x + 2) (5x + 4) 15x2 + 12x + 10x Multiplying by FOIL I ---> Inner Terms 15x2 + 12x + 10x

F O I L (3x + 2) (5x + 4) 15x2 + 12x + 10x + 8 Multiplying by FOIL L ---> Last Terms 15x2 + 12x + 10x + 8

F O I L (3x + 2) (5x + 4) 15x2 + 22x + 8 15x2 + 12x + 10x + 8 Multiplying by FOIL (3x + 2) (5x + 4) F O I L Combine Like Terms 15x2 + 22x + 8 15x2 + 12x + 10x + 8

Multiplying by FOIL (3x + 2) (5x + 4) F O I L 15x2 + 22x + 8

Multiplying by FOIL (5x + 1) (5x + 1) F O I L

F O I L (5x + 1) (5x + 1) 25x2 Multiplying by FOIL F ---> First Terms

F O I L (5x + 1) (5x + 1) 25x2 + 5x Multiplying by FOIL O ---> Outer Terms 25x2 + 5x

F O I L (5x + 1) (5x + 1) 25x2 + 5x + 5x Multiplying by FOIL I ---> Inner Terms 25x2 + 5x + 5x

F O I L (5x + 1) (5x + 1) 25x2 + 5x + 5x + 1 Multiplying by FOIL L ---> Last Terms 25x2 + 5x + 5x + 1

F O I L (5x + 1) (5x + 1) 25x2 + 10x + 1 25x2 + 5x + 5x + 1 Multiplying by FOIL (5x + 1) (5x + 1) F O I L Combine Like Terms 25x2 + 10x + 1 25x2 + 5x + 5x + 1

Multiplying by FOIL (5x + 1) (5x + 1) F O I L 25x2 + 10x + 1

Multiplying by FOIL (5x + 1)2 (5x + 1) (5x + 1) 25x2 + 10x + 1

Special Products In the process of using the FOIL method on products of certain types of binomials, we see specific patterns that lead to special products. Square of a Binomial (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2 Product of the Sum and Difference of Two Terms (a + b)(a – b) = a2 – b2

Special Products Although you will arrive at the same results for the special products by using the techniques of this section or last section, memorizing these products can save you some time in multiplying polynomials.

Evaluating Polynomials We can use function notation to represent polynomials. For example, P(x) = 2x3 – 3x + 4. Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved. Example: Find the value P(2) = 2x3 – 3x + 4. P(2) = 2(2)3 – 3(2) + 4 = 2(8) + 6 + 4 = 6

Evaluating Polynomials Techniques of multiplying polynomials are often useful when evaluating polynomial functions at polynomial values. Example: If f(x) = 2x2 + 3x – 4, find f(a + 3). We replace the variable x with a + 3 in the polynomial function. f(a + 3) = 2(a + 3)2 + 3(a + 3) – 4 = 2(a2 + 6a + 9) + 3a + 9 – 4 = 2a2 + 12a + 18 + 3a + 9 – 4 = 2a2 + 15a + 23

Homework 5.3 #3-45 multiplies odd & 71-77 odd 5.4 #1-33 odd & 49, 53, 55, 61, 65

SUMMARY Can You?... 1) multiply monomials and polynomials. 2) multiply special types of binomials using the FOIL method.

The Power Rule The Power Rule and Power of a Product or Quotient Rule for Exponents If a and b are real numbers and m and n are integers, then (am)n = amn Power Rule (ab)n = an · bn Power of a Product Power of a Quotient

The Power Rule Example: Simplify each of the following expressions. (23)3 = 23·3 = 29 = 512 (x4)2 = x4·2 = x8 (5x2y)3 = 53 · (x2)3 · y3 = 125x6 y3

Summary of Exponent Rules If m and n are integers and a and b are real numbers, then: Product Rule for exponents am · an = am+n Power Rule for exponents (am)n = amn Power of a Product (ab)n = an · bn Power of a Quotient Quotient Rule for exponents Zero exponent a0 = 1, a  0 Negative exponent

Simplifying Expressions Simplify by writing the following expression with positive exponents or calculating. Power of a quotient rule Power of a product rule Power rule for exponents Negative exponents Negative exponents Quotient rule for exponents

Operations with Scientific Notation Multiplying and dividing with numbers written in scientific notation involves using properties of exponents. Example Perform the following operations. (7.3  102)(8.1  105) 1) = (7.3 · 8.1)  (102 · 105) = 59.13  103 = 59,130 2)

Operations with Scientific Notation Multiplying and dividing with numbers written in scientific notation involves using properties of exponents. Example Perform the following operations. (7.3  102)(8.1  105) 1) = (7.3 · 8.1)  (10-2 ·105) = 59.13  103 = 59,130 2)