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University of Phoenix MTH 209 Algebra II

University of Phoenix MTH 209 Algebra II. The FUN continues!. Chapter 4. What we are doing now is using all we have used so far and add a few complications like the concept of a root (square root, cube root, etc.). The big word for MTH 209 is a polynomial (poly = parrot)

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University of Phoenix MTH 209 Algebra II

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  1. University of PhoenixMTH 209 Algebra II The FUN continues!

  2. Chapter 4 • What we are doing now is using all we have used so far and add a few complications like the concept of a root (square root, cube root, etc.). • The big word for MTH 209 is a polynomial (poly = parrot) • (no… poly = many, nomial = number)

  3. Section 4.1 • Remember the Product and Quotient Rules am * an = am+n And the zero exponent and a0 = 1

  4. Taking what they throw at ya: Ex 1 page 256 • a) 23· 22 = 23+2 = 25 = 32 (calculator!) • b) x2 · x4 · x= x2 · x4 · x1=x7 • c) 2y3 ·4y8=(2)(4)y3y8=8y11 • d) -4a2b3(-3a5b9)=(-4)(-3) a2a5b3b9=12a7b12 **Ex. 7-18**

  5. Zero Exponent Example 2 page 257 a) 50 = 1 b) (3xy)0 = 1 c) b0·b9=1 ·b9 c) 20+30 = 1+1=2 **Ex. 19-28**

  6. Section 4.1 • Remember the Quotient Rules If then If n>m, then

  7. More thrown at ya: Ex 3 page 258 • a) x7/x4= x7-4= x3 • b) w5/w3=w5-3=w2 • c) • d) **Ex. 29-40**

  8. What about raising an exponent to and exponent? • Yes, we have to try to break it if we can! • What ABOUT raising an exponent to a power? • You MULTIPLY the exponents!

  9. Power of a Power • (w4)3 = w4 w4 w4= w12 • Which is the same as w4*3 = w12 • Definition time:The Power Rule (am)n = amn

  10. Example 4 page 259 • a) (23)8 = 23·8=224 • b) (x2)5=x2·5=x10 • c) 3x8(x3)6= 3x8(x3·6) = 3x8(x18) = 3x8+18=3x26 • d) **Ex. 41-50**

  11. Now, to the power of a product. What if there is more than just x inside the ()’s? • For example: (2x)3=2x2x2x = 2*2*2*x*x*x= 23x3 (then do the math and get) 8x3 So the Power of a Product Rule (ab)n = anbn

  12. Example 5 page 259 • a) (-2x)3=(-2)3x3=-8x3 • b) (-3a2)4= (-3)4 (a2)4 = 81a8 • c) (5x3y2)3 = 53(x3)(y2)3=124x9y6 **Ex. 51-58**

  13. The power of a quotient • It is JUST what what you’d expect! • Raise the top and the bottom to the power by themselves, then work with it! For example

  14. The Power of a Quotient • Where b does NOT equal 0

  15. Example 6 • a) • b) • c) **Ex. 59-66**

  16. All summed up! • Look to page 261 of your text for ALL these power rules to date summarized. • Use these for quizzes and tests!

  17. Section 4.1 With your OWN hand • Definitions Q1-6 • Product rule Q7-18 • Zero exponents Q19-28 • Quotient Rule Q29-40 • Power Rule Q41-50 • Power of Product Q51-58 • Power of Quotient Rule Q59-66 • Simplify random stuff Q67-88 • Wordy Problems Q89-96 Remember… tan lines are ones with homework or group work problems in them

  18. Section 4.2 Negative exponents (I’ve let you see them already- don’t tell anyone) • 1/x is the same as x-1 • So Negative Integral Exponents are defined as

  19. What is one again? • a-n * an = a-n+n = a0 = 1

  20. Example 1 page 265 • a) • b) • c)-9-2=-(9-2)= -1/92=-1/81 • d) **Ex. 7-16**

  21. Some pitfalls • Watch the negative sign… • -5-2 = -(5-2) so the answer is –1/25 • Also if you see this… make it simpler

  22. Helpful rules (pg 260) with negative powers… • a must be non-zero

  23. Example 2 page 266 • a) • b) • c) • d) **Ex.17-26**

  24. Rules for Integral Exponents • Just like positive exponents, if you multiply the numbers, add the exponents! • x-2 * x-3 = x-2+(-3) = x-5 • Or with division…

  25. See Page 267 for MORE summary rules!

  26. Example 3 Working it out… page 268 • a) b-3b5= b5-3=b2 • b) -3x-3 * 5x2 =-15x2-3= -15x-1 = -15/x • c) • d) **Ex. 27-42**

  27. Example 4 page 268 • a) (a-3)2=a-3*2=a-6 • b) (10x-3)-2=10-2(x-3)-2 = 10-2x6= x6/100 • c) **Ex. 43-58**

  28. Scientific Notation… or “how I learned to love large numbers” • We look at big numbers like 100000. and need to tell people later how many zeros we took out (who wants to write that many zeros anyway?). • 1.00000 we ‘hopped’ the decimal place 5 hops to the left (positive) direction • We write this as 1 x 105

  29. Small numbers… • 0.0000000003 • We need to hop the decimal place to the right (negative direction) and place it to the right of the first integer. • 0x0000000003. We took 10 hops to the right • We code it as 3 x 10-10

  30. nice examples • 10(5.32) = 53.2 • 102(5.32)=100(5.32)=532 • 103(5.32)=1000(5.32)=5320 and • 10-1(5.32)=.1(5.32)=.532 • 10-2(5.32)=.01(5.32)=.0532 • 10-3(5.32)=.001(5.32)=.00532

  31. Example 6 page 270 • Write in standard notation • a) 7.02 x 106 = 7020000 = 7,020,000 • b) 8.13 x 10-5 = .0000813 **Ex. 65-72**

  32. Example 7 page 271 • Write in Scientific Notation a) 7346200 it’s bigger than 10 so the exponent will be positive 7.3462 x 106 • 0.0000348 it’s less than 10 so the exponent will be negative 3.48 x 10-5 c) 135 x 10-12 it should start with 1.35 so we need to go positive 2 places changing it to 1.35x10-10 **Ex. 73-80**

  33. Example 8 Computing with scientific notation (more exponents) page 272 • a) (3x106)(2x108) = 3*2*106*108=6x1014 • b)

  34. Example 8c • (5x10-7)3=53(10-7)3=125 (10-21)=1.25(102)(10-21)= 1.25 x 10-19 **Ex. 81-92**

  35. Example 9 page 272First Sci. Note. then math • a) (3,000,000)(0.0002) =3x106·2x10-4 = 6x102 b) (20,000,000)3(0.0000003) =8x1021 ·3x10-7 = 24x1014 need to move decimal 24. to 2.4 which is one to the right, or bigger! =24x1015**Ex. 93-100**

  36. More pen to paper Section 4.2 • Definitions Q1-6 • Get rid of negative exponents Q7-26 • Write numbers in standard notation Q27-58 • Present Value Formula Q59-64 • Scientific Notation Q65-80 • Computations with S.N. Q81-108 • Word problems Q109-116 (Learning Team)

  37. Section 4.3 • Polynomials…You have already seen them in Chapter 1 and 2 so don’t panic!

  38. Poly-want a cracker? • What is a term? • 4x3 • -x2y3 • 6ab • -2 • xyza3

  39. A polynomial is a set of those terms • A scrapbook of polynomials: • x2+5x+3 • 4x3+10x2+2x+100 • x+4

  40. Simplify – the standard way • 4x3+x+(-15x2)+(-2) • We like to get rid of ( )’s • We like them in order of decreasing exponent and alphabetized if possible • The above becomes 4x3-15x2+x-2 • a2b + b2a + b3a2+ a3b2 simplifies to… • a3b2 + b3a2 + a2b + b2a

  41. The degree of the polynomial • We label the terms by the number in the exponent • 4x3-15x2+x-2 • 3rd order term (or degree term) • 2nd order term (or degree term) • 1st order term (or degree term) • 0th order term (or degree term) this isalso called a constant [Try this on a calculator… if the power of the x with the constant is zero, then it is x0

  42. You are constantly… • [Try this on a calculator… if the power of the x with the constant is zero, then it is x0 • What is 60? Or 10? Or even 10000? • So what is 10 * x0 ?

  43. What’s in a coefficient? • 4x3-15x2+x-2 • The third order term’s coefficient is 4 • The second order term’s coefficient is –15 • The first order term’s coefficient is 1 • The constant is –2 (the coefficient to the zeroth order term)

  44. Example 1 Got Coefficient? pg277 • What are the coefficients of x3 and x2 in each: • a) x3+5x2-6 which is _x3+5x2-6 • 1 and 5 • b) 4x6- x3 + x is the same as 4x6- x3 + 0x2 + x • So it’s –1 and 0 **Ex. 7-12**

  45. Again… how we like to order them • We don’t like -4x2+1+5x+x3 • We do like x3-4x2+ 5x+ 1 • The coefficient of the highest order term (x3) is called the leading coefficient • The leading coefficient is 1 in this case • The order of the polynomial is the exponent (or power) of the HIGHEST term

  46. Special Definitions • A monomial has only one term • x, x2 , 3 • A binomial has two terms • x+5 , or x3+4 • A trinomial has three terms (see a trend?) • 10x4+6x+100

  47. Example 2 pg 277 • Identify each as a polynomial as either a monomial, binomial or trinomial and state it’s degree… • a) 5x2-7x3+2  is a trinomial of 3rd order (degree) • b) x43-x2 is a binomial of 43rd order (degree) • c) 5x=5x1 is a polynomial of degree 1 (order) • d) –12  is a monomial of degree 0 (order) • **Ex. 13-24**

  48. Example 3 plug in the numberpg 278 • The value of a polynomial… • a) Find the value of –3x4-x3+20x+3 when x=1 -3(1)4-(1)3+20(1)+3 = -3-1+20+3 = 19 • b) The same equation when x=-2  -3(-2)4-(-2)3+20(-2)+3 = -48+8-40+3 = -77 **Ex. 25-32**

  49. Example 4 another look pg279 a) and if P(x) = –3x4-x3+20x+3 and you’re told to find P(1) It’s the same as 3a) -3(1)4-(1)3+20(1)+3 = -3-1+20+3 = 19 b) If D(a)=a3-5 find D(0), D(1),D(2) 03-5=-5, 13-5=1-5=-4, 23-5=8-5=3 **Ex. 33-38**

  50. Adding Polynomials • To add two polynomials, add the like terms • (You have done all this already as well!)

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