Chapter 8-Polynomials

8.1-Multiplying Monomials Monomial-a number, a variable, or the product of a number and one or more variables. –Example: –-5 –3a –a2b3–a2b3 Constants-monomials that are real numbers –A number by itself, without a variable (Ex: 4)

8.1-Multiplying Monomials To MULTIPLY powers that have the SAME BASE, just simply ADD the exponents and leave the base the same. Example: 2 3 * 2 5 = 2 8 x 5 * x = x 6 (x is the same as x 1 and = 6)

Simplify 1. (-7c 3 d 4 ) (4cd 3 ) = -28c 4 d 7 2.(5a 2 b 3 c 4 ) (6a 3 b 4 c 2 ) = 30a 5 b 7 c 6

Find the Power of a Power To find the power of a power, multiply the exponents. Example: (2 2 ) 3 = 2 6

Simplify 1.(p 3 ) 5 = p 15 2.[(3 2 ) 4 ] 2 = 3 16

8.2-Dividing Monomials

Dividing Powers with the Same Base To DIVIDE powers that have the SAME BASE, SUBTRACT the exponents. Example: c4c4 c2c2 Answer: c 2

Power of Zero & Negative Exponents Anytime you have a number (except zero) to the power of ZERO, the answer is ONE. Example: 3 0 = 1 Anytime you have a negative exponent, flip it to make is positive. Example: 4 -2 = Example: 1 =

Simplify 1. #15 on page 421 Answer: #17 on page 421 Answer: y 2 z 7 3.# 29 on page 421 Answer: 6k 17 h 3

8.3-Scientific Notation

What is Scientific Notation? Scientific Notation-a number written as a product of a factor and a power of 10. –The factor must be greater than or equal to 1 and less than 10. –Used for numbers that are very large or very small.

Express Numbers in Standard Notation Express the number in standard notation: 2.45 x 10 8 The power of 8 means to move the decimal 8 places to the right and fill in any empty holes with zeroes. Answer: 245,000,000

Express Numbers in Standard Notation Express the number in standard notation: 3 x The power of –5 means to move the decimal 5 places to the left and fill in any empty holes with zeroes. Answer:.00003

Express Numbers in Scientific Notation Scientific Notation is the reverse of Standard Notation Express the number in scientific notation. 30,500,000 Put the decimal right after the first number That gives you

How many places do you go to get the decimal back to where it was in the beginning? 7….to the right! So the answer is 3.05 x 10 7

Express the number in scientific notation Put the decimal after the first number that is not zero. That gives you How many places do you go to get the decimal back to where it was in the beginning? 4….to the left! So the answer is 7.81 x 10 -4

Multiplication and Division with Scientific Notation Example: (5 x )(2.9 x 10 2 ) Just simply type it in your calculator. For the times ten (x10) you will use E. (2 nd, comma) In your calculator you should see…. (5E-8)(2.9E2) Press ENTER to get your answer.

Assignment Page 837 Section 8.2 and 8.3 All

8.4-Polynomials

Polynomials A polynomial is a monomial or a sum of monomials. Types of polynomials –Binomial: sum or difference of two monomials –Trinomial: sum or difference of three monomials.

Degrees Degree of a monomial-the sum of the exponents Example: the degree of 8y 4 is 4, the degree of 2xy 2 z 3 is 6 (because if you add all the exponents of the variables you get 6)

Degrees Degree of a polynomial-the greatest degree of any term in the polynomial –Find the degree of each term, the highest is the degree of the polynomial Example: 4x 2 y 2 + 3x Find the degree of each term 4x 2 y 2 has a degree 4 3x 2 has a degree of 2 5 has no degree The greatest is 4, so that’s the degree of the polynomial.

Arrange Polynomials Arrange Polynomials in ascending or descending order Ascending-least to greatest Descending-greatest to least Example: 6x 3 –12 + 5x in descending order. 6x 3 + 5x –12

8.5-Adding and Subtracting Polynomials

When adding or subtracting polynomials remember to combine LIKE TERMS. Example: (3x 2 – 4x + 8) + (2x – 7x 2 – 5) Notice which terms are alike…combine these terms. (They have been color coded) 3x 2 – 7x 2 = -4x 2 – 4x + 2x = -2x 8 – 5 = 3 So the answer is… -4x 2 - 2x + 3 Be sure to put the powers in descending order.

Add or Subtract Polynomials 1.(5y 2 – 3y + 8) + (4y 2 – 9) Answer: 9y 2 –3y –1 2.(3ax 2 – 5x – 3a) – (6a – 8a 2 x + 4x) Answer: 3ax 2 – 9x – 9a + 8a 2 x

8.6-Multiplying a Polynomial by a Monomial

Examples 1. -2x 2 (3x 2 – 7x + 10) Notice the –2x 2 on the outside of the parenthesis……you must distribute this. -2x 2 * 3x 2 = -6x 4 -2x 2 * -7x = 14x 3 -2x 2 * 10 = -20x 2 Answer: -6x x 3 – 20x 2

Examples 2. 4(3d 2 + 5d) – d(d 2 –7d + 12) Notice you have to distribute the 4 and –d 4 * 3d 2 = 12d 2 4 * 5d = 20d -d * d 2 = -d 3 -d * -7d = 7d 2 -d * 12 = -12d Put it all together…. 12d d –d 3 + 7d 2 – 12d Notice the like terms…. Answer: -d d 2 + 8d

Examples 3. y(y – 12) + y(y + 2) + 25 = 2y(y + 5) – 15 Distribute y, y and 2y y * y = y 2 y * -12 = -12y y * y = y 2 y * 2 = 2y Don’t forget the +25 2y * y = 2y 2 2y * 5 = 10y Don’t forget the -15

Now you have……. y 2 – 12y + y 2 + 2y + 25 = 2y y –15 Combine like terms…. 2y 2 –10y + 25 = 2y y – 15 Now you have to solve because you have an equals sign Answer: y = 2

8.7-Multiplying Polynomials

Multiplying Two Binomials Example: (x + 3) (x + 2) This can be done a number of ways. Use either FOIL or Box Method

FOIL (x + 3) (x + 2) F-Multiply the First terms in each x * x = x 2 O-Multiply the Outer terms x * 2 = 2x I-Multiply the Inner terms 3 * x = 3x L-Multiply the Last terms 3 * 2 = 6 Answer: x 2 + 5x + 6

Box Method

Combine Combine the two that are circled Answer: x 2 + 5x + 6

Polynomials (4x + 9) (2x 2 – 5x + 3) Multiply 4x by (2x 2 –5x + 3) 4x * 2x 2 = 8x 3 4x * -5x = -20x 2 4x * 3 = 12x Multiply 9 by (2x 2 –5x + 3) 9 * 2x 2 = 18x 2 9 * -5x = -45x 9 * 3 = 27

Or…BOX METHOD Answer: 8x 3 –2x 2 –33x + 27