Imagine how weird phones would look if your mouth was nowhere near your ears.

FST Section 9.1

 Find the volume of an open box with one edge of unknown length, a second edge of one unit longer, and a third edge one unit shorter.  Hint: draw a picture  Hint: write as a polynomial function

 Monomial = a number or product of numbers and variables with whole number exponents  Polynomial = a monomial, or a sum or difference of monomials.

The degree is 3 The degree is 2

 Standard Form = the terms of a polynomial are ordered from left to right in descending order, which means from the greatest exponent to the least.  The leading coefficient is the coefficient of the term with highest degree.  Examples: Write each polynomial in standard form.

PolynomialsNot Polynomials |2b 3 – 6b|

 Naming Polynomials by the Number of Terms and by Degree PolynomialDegreeName by Degree # of Terms Name by # of Terms 120Constant1Monomial 8x1Linear1Monomial 2Quadratic2Binomial 3Cubic2Binomial 2Quadratic3Trinomial 4Quartic4Polynomial

Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial. Example 2: Classifying Polynomials A. 3 – 5x 2 + 4xB. 3x 2 – 4 + 8x 4 –5x 2 + 4x + 3 Write terms in descending order by degree. Leading coefficient: –5 Terms: 3 Name: quadratic trinomial Degree: 2 8x 4 + 3x 2 – 4 Write terms in descending order by degree. Leading coefficient: 8 Terms: 3 Name: quartic trinomial Degree: 4

Check It Out! Example 2 Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial. a. 4x – 2x 2 + 2b. –18x 2 + x 3 – 5 + 2x –2x 2 + 4x + 2 Leading coefficient: –2 Terms: 3 Name: quadratic trinomial Degree: 2 1x 3 – 18x 2 + 2x – 5 Leading coefficient: 1 Terms: 4 Name: cubic polynomial with 4 terms Degree: 3

To add or subtract polynomials, combine like terms. You can add or subtract horizontally or vertically.

Example 3: Adding and Subtracting Polynomials Add or subtract. Write your answer in standard form. A. (2x – x) + (5x x + x 3 ) (2x – x) + (5x x + x 3 ) Add vertically. Write in standard form. Align like terms. Add. 2x 3 – x + 9 +x 3 + 5x 2 + 7x + 4 3x 3 + 5x 2 + 6x + 13

Example 3: Adding and Subtracting Polynomials Add or subtract. Write your answer in standard form. B. (3 – 2x 2 ) – (x – x) (3 – 2x 2 ) – (x – x) Add the opposite horizontally. Write in standard form. Group like terms. Add. (–2x 2 + 3) + (–x 2 + x – 6) (–2x 2 – x 2 ) + (x) + (3 – 6) –3x 2 + x – 3

Check It Out! Example 3a Add or subtract. Write your answer in standard form. (–36x 2 + 6x – 11) + (6x x 3 – 5) 16x 3 – 30x 2 + 6x – 16 (5x x 2 ) – (15x 2 + 3x – 2) 5x 3 – 9x 2 – 3x + 14

To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents.

Find each product. A. 4y 2 (y 2 + 3) Distribute. B. fg(f 4 + 2f 3 g – 3f 2 g 2 + fg 3 ) 4y 2  y 2 + 4y 2  3 Multiply. 4y y 2 Distribute. Multiply. fg  f 4 + fg  2f 3 g – fg  3f 2 g 2 + fg  fg 3 f 5 g + 2f 4 g 2 – 3f 3 g 3 + f 2 g 4

Find each product. a. 3cd 2 (4c 2 d – 6cd + 14cd 2 ) b. x 2 y(6y 3 + y 2 – 28y + 30) 3cd 2  4c 2 d – 3cd 2  6cd + 3cd 2  14cd 2 12c 3 d 3 – 18c 2 d c 2 d 4 x 2 y  6y 3 + x 2 y  y 2 – x 2 y  28y + x 2 y  30 6x 2 y 4 + x 2 y 3 – 28x 2 y x 2 y

To multiply any two polynomials, use the Box Method to ensure that each term in the second polynomial is multiplied by each term in the first. Keep in mind that if one polynomial has m terms and the other has n terms, then the product has mn terms before it is simplified.

Find the product. (a – 3)(2 – 5a + a 2 ) Step 1: Write polynomials in standard form. Step 2: Create a box with a row/column for each term. Step 3: Multiply Step 4: Combine like terms

(x 2 – 7x + 5)(x 2 – x – 3) Find the product. Step 1: Write polynomials in standard form. Step 2: Create a box with a row/column for each term. Step 3: Multiply Step 4: Combine like terms

Find the product. Step 1: Write polynomials in standard form. Step 2: Create a box with a row/column for each term. Step 3: Multiply Step 4: Combine like terms (x 2 – 4x + 1)(x 2 + 5x – 2)

Find the product. (a + b) 2 (a + b) 3

There are n+1 terms and the powers sum for each term = n.

Expand each expression. A. (k – 5) Identify the coefficients for n = 3, or row 3. [1(k) 3 (–5) 0 ] + [3(k) 2 (–5) 1 ] + [3(k) 1 (–5) 2 ] + [1(k) 0 (–5) 3 ] k 3 – 15k k – 125

Expand each expression. a. (x + 2) Identify the coefficients for n = 3, or row 3. [1(x) 3 (2) 0 ] + [3(x) 2 (2) 1 ] + [3(x) 1 (2) 2 ] + [1(x) 0 (2) 3 ] x 3 + 6x x + 8

b. (x – 4) Identify the coefficients for n = 5, or row 5. [1(x) 5 (–4) 0 ] + [5(x) 4 (–4) 1 ] + [10(x) 3 (–4) 2 ] + [10(x) 2 (–4) 3 ] + [5(x) 1 (–4) 4 ] + [1(x) 0 (–4) 5 ] x 5 – 20x x 3 – 640x x – 1024

Expand each expression. c. (y - 3) 4 d. (4x + 5) 3 y 4 – 12y y 2 – 108y x x x + 125

 The general form for a polynomial in x is: n: degree of the polynomial a: the coefficients a n : leading coefficient The leading coefficient MUST be attached to the variable with the HIGHEST degree!

 Degree of polynomial  General shape  Even or odd function  End behavior  Number of possible “solutions” (x-intercepts)  Always one y-intercept  Number/location of possible Maximum points and minimum points ▪ Absolute versus relative (global versus local)

 Analysis:  y = a n x n format  Degree = n ▪ Odd or even? (only odd or only even terms) ▪ End behavior ▪ Max. number of possible x-intercepts ▪ Absolute versus local mins/maxes ▪ other

 A company has pieces of cardboard that are 60 cm by 80 cm. They want to make boxes from them (without a top) to hold equipment. Squares with sides of length x are cut from each corner and the resulting flaps are folded to make an open box.

 What’s the (max? min?) volume of the box that can be created?  Draw a picture to help!!

 Look at our answer for the cardboard box:  What’s the degree of the polynomial?  What’s the leading coefficient?  To standardize, we ALWAYS want to write polynomials in standard form (terms in order of descending degree)

 Use the volume formula we got from the cardboard box.  Find V(10)  What does this mean/represent?

 Monomial: a polynomial with one term  Binomial: a polynomial with two terms  Trinomial: a polynomial with three terms  Polynomial: general term for any polynomial (especially if it has more than three terms)

 Polynomials may have more than one variable  For example: x 2 y 3 – 3y 2 + 2x 2 – 6 This would be a polynomial in x and y  The degree of a polynomial in more than one variable is the largest sum of the exponents of the variables in any term.  In the example above, the degree is 5.

 Express the surface area and the volume of a cube with sides of length (a+b) in terms of a and b.  State the degree of each polynomial.

 Tamara is saving her summer earnings for college. The table shows the amount of money saved each summer.  At the end of each summer, she put her money in a savings account with an annual yield of 7%. How much will be in her account when she goes to college, if no additional money is added or withdrawn, and the interest rate remains constant? After Grade Amount Saved 8$600 9$900 10$ $ $1600

 At the end of each summer, she put her money in a savings account with an annual yield of 7%. How much will be in her account when she goes to college, if no additional money is added or withdrawn, and the interest rate remains constant?  Can you come up with a polynomial model?? (hint: start with an “exponential concept”) After Grade Amount Saved 8$600 9$900 10$ $ $1600

 Page : # 1 – 6, 9 – 12, 16  Define the following terms: functionlinear model independent variableexponential model dependent variablequadratic model mathematical model  Provide two examples of functions you use or see in everyday life. Describe the independent and dependent variables and how the functions are used.