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Review Topics (Chapter 0 & 1)

This review covers topics such as exponents, radical expressions, factoring, quadratic equations, rational expressions, solving inequalities, and more. Learn and practice various exponent rules and simplify radical expressions. Understand how to add, subtract, multiply, and divide polynomials. Also, learn how to solve polynomial equations by factoring.

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Review Topics (Chapter 0 & 1)

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  1. Review Topics (Chapter 0 & 1) • Exponents & Radical Expressions • Factoring • Quadratic Equations • Rational Expressions • Rational Equations & Clearing Fractions • Radical Equations • Solving Inequalities • Linear Graphing and Functions • Function Evaluation • Slope and Average Rate of Change • Difference Quotient

  2. Review of Exponents 82 =8 • 8 = 64 24 = 2 • 2 • 2 • 2 = 16 x2 = x • x x4 = x • x • x • x Base = x Base = x Exponent = 2 Exponent = 4 Exponents of 1 Zero Exponents Anything to the 1 power is itself Anything to the zero power = 1 51 = 5 x1 = x (xy)1 = xy 50 = 1 x0 = 1 (xy)0 = 1 Negative Exponents 5-2 = 1/(52) = 1/25 x-2 = 1/(x2) xy-3 = x/(y3) (xy)-3 = 1/(xy)3 = 1/(x3y3) a-n = 1/an 1/a-n = an a-n/a-m = am/an

  3. Raising Quotients to Powers a n b an bn a -n b a-n b-n bn an b n a = = = = Examples: 3 2 32 9 4 42 16 = = 2x 3 (2x)3 8x3 y y3 y3 = = 2x -3 (2x)-3 1 y3 y3 y y-3 y-3(2x)3 (2x)3 8x3 = = = =

  4. Product Rule am• an = a(m+n) x3• x5 = xxx • xxxxx = x8 x-3 • x5 = xxxxx = x2 = x2 xxx 1 x4 y3 x-3 y6 = xxxx•yyy•yyyyyy = xy9 xxx 3x2 y4 x-5• 7x = 3xxyyyy • 7x = 21x-2 y4 = 21y4 xxxxx x2

  5. Quotient Rule am = a(m-n) an 43= 4 • 4 • 4 = 41 = 4 43 = 64 = 8 = 4 42 4 • 4 42 16 2 x5 = xxxxx = x3x5 = x(5-2) = x3 x2 xx x2 15x2y3 = 15 xx yyy = 3y215x2y3 = 3 • x -2 • y2 = 3y2 5x4y 5 xxxx y x2 5x4y x2 3a-2 b5 = 3 bbbbb bbb = b83a-2 b5 = a(-2-4)b(5-(-3)) = a-6 b8 = b8 9a4b-3 9aaaa aa 3a6 9a4b-3 3 3 3a6

  6. Powers to Powers (am)n = amn (a2)3 a2• a2 • a2 = aa aa aa = a6 (24)-2 = 1 = 1 = 1 = 1/256 (24)2 24• 24 16 • 16 28 256 (x3)-2 = x –6 = x 10 = x4 (x -5)2 x –10 x 6 (24)-2 = 2-8 = 1 = 1

  7. Products to Powers (ab)n = anbn (6y)2 = 62y2 = 36y2 (2a2b-3)2 = 22a4b-6 = 4a4 = a 4(ab3)3 4a3b9 4a3b9b6 b15 What about this problem? 5.2 x 1014 = 5.2/3.8 x 109 1.37 x 109 3.8 x 105 Do you know how to do exponents on the calculator?

  8. Square Roots & Cube Roots A number b is a cube root of a number a if b3 = a 8 = 2 since 23 = 8 Notice that 8 breaks down into 2 • 2 • 2 So, 8 =  2 • 2 • 2 See a ‘group of 3’ –> bring it outside the radical (the cube root sign) Example: 200 = 2 • 100 = 2 • 10 • 10 = 2 • 5 • 2 • 5 • 2 = 2 • 2 • 2 • 5 • 5 = 2 25 A number b is a square root of a number a if b2 = a 25 = 5 since 52 = 25 Notice that 25 breaks down into 5 • 5 So, 25 =  5 • 5 See a ‘group of 2’ -> bring it outside the radical (square root sign). Example: 200 = 2 • 100 = 2 • 10 • 10 = 10 2 3 3 3 3 3 3 3 3 Note: -25 is not a real number since no number multiplied by itself will be negative Note: -8 IS a real number (-2) since -2 • -2 • -2 = -8 3

  9.     Nth Root ‘Sign’ Examples Even radicals of positive numbers Have 2 roots. The principal root Is positive. 16 = 4 or -4 not a real number -16 Even radicals of negative numbers Are not real numbers. 4 -16 not a real number Odd radicals of negative numbers Have 1 negative root. 5 -32 = -2 5 Odd radicals of positive numbers Have 1 positive root. 32 = 2

  10. m Xm Ym X Y = Exponent Rules (XY)m = xmym

  11. Examples to Work through

  12. Some Rules for Simplifying Radical Expressions

  13. Practice Problems

  14. Operations on Radical Expressions • Addition and Subtraction (Combining LIKE Terms) • Multiplication and Division • Rationalizing the Denominator

  15. Radical Operations with Numbers

  16. Multiplying Radicals (FOIL works with Radicals Too!)

  17. Rationalizing the Denominator • Remove all radicals from the denominator

  18. Adding & Subtracting Polynomials Combine Like Terms (2x2 –3x +7) + (3x2 + 4x – 2) = 5x2 + x + 5 (5x2 –6x + 1) – (-5x2 + 3x – 5) = (5x2 –6x + 1) + (5x2 - 3x + 5) = 10x2 – 9x + 6 Types of Polynomials f(x) = 3 Degree 0 Constant Function f(x) = 5x –3 Degree 1 Linear f(x) = x2 –2x –1 Degree 2 Quadratic f(x) = 3x3 + 2x2 – 6 Degree 3 Cubic

  19. Multiplication of Polynomials Step 1: Using the distributive property, multiply every term in the 1st polynomial by every term in the 2nd polynomial Step 2: Combine Like Terms Step 3: Place in Decreasing Order of Exponent 4x2 (2x3 + 10x2 – 2x – 5) = 8x5 + 40x4 –8x3 –20x2 (x + 5) (2x3 + 10x2 – 2x – 5) = 2x4 + 10x3 – 2x2 – 5x + 10x3 + 50x2 – 10x – 25 = 2x4 + 20x3 + 48x2 –15x -25

  20. Binomial Multiplication with FOIL (2x + 3) (x - 7) F. O. I. L. (First) (Outside) (Inside) (Last) (2x)(x) (2x)(-7) (3)(x) (3)(-7) 2x2 -14x 3x -21 2x2 -14x + 3x -21 2x2 - 11x -21

  21. Division by a Monomial 3x2 + x 5x3 – 15x2 x 15x 4x2 + 8x – 12 5x2y + 10xy2 4x2 5xy 15A2 – 8A2 + 12 12A5 – 8A2 + 12 4A 4A

  22. Review: Factoring Polynomials To factor a number such as 10, find out ‘what times what’ = 10 10 = 5(2) To factor a polynomial, follow a similar process. Factor: 3x4 – 9x3 +12x2 3x2 (x2 – 3x + 4) Another Example: Factor 2x(x + 1) + 3 (x + 1) (x + 1)(2x + 3)

  23. Solving Polynomial Equations By Factoring Zero Product Property : If AB = 0 then A = 0 or B = 0 Solve the Equation: 2x2 + x = 0 Step 1: Factor x (2x + 1) = 0 Step 2: Zero Product x = 0 or 2x + 1 = 0 Step 3: Solve for X x = 0 or x = - ½ Question: Why are there 2 values for x???

  24. Factoring Trinomials To factor a trinomial means to find 2 binomials whose product gives you the trinomial back again. Consider the expression: x2 – 7x + 10 (x – 5) (x – 2) The factored form is: Using FOIL, you can multiply the 2 binomials and see that the product gives you the original trinomial expression. How to find the factors of a trinomial: Step 1: Write down 2 parentheses pairs. Step 2: Do the FIRSTS Step3 : Do the SIGNS Step4: Generate factor pairs for LASTS Step5: Use trial and error and check with FOIL

  25. Practice • Factor: • y2 + 7y –30 4. –15a2 –70a + 120 • 10x2 +3x –18 5. 3m4 + 6m3 –27m2 • 8k2 + 34k +35 6. x2 + 10x + 25

  26. Special Types of Factoring Square Minus a Square A2 – B2 = (A + B) (A – B) Cube minus Cube and Cube plus a Cube (A3 – B3) = (A – B) (A2 + AB + B2) (A3 + B3) = (A + B) (A2 - AB + B2) Perfect Squares A2 + 2AB + B2 = (A + B)2 A2 – 2AB + B2 = (A – B)2

  27. Quadratic Equations • General Form of Quadratic Equation • ax2 + bx + c = 0 • a, b, c are real numbers & a 0 • A quadratic Equation: x2 – 7x + 10 = 0 a = _____ b = _____ c = ______ • Methods & Tools for Solving Quadratic Equations • Factor • Apply zero product principle (If AB = 0 then A = 0 or B = 0) • Square root method • Completing the Square • Quadratic Formula 1 -7 10 Example1:Example 2: x2 – 7x + 10 = 0 4x2 – 2x = 0 (x – 5) (x – 2) = 0 2x (2x –1) = 0 x – 5 = 0 or x – 2 = 0 2x=0 or 2x-1=0 + 5 + 5 + 2 + 2 2 2 +1 +1 2x=1 x = 5 or x = 2 x = 0 or x=1/2

  28. Square Root Method If u2 = d then u = d or u = - d. If u2 = d then u = + d • Solving a Quadratic Equation with the Square Root Method • Example 1: Example 2: • 4x2 = 20 (x – 2)2 = 6 • 4 • x – 2 = +6 • x2 = 5 + 2 + 2 • x = + 5 x = 2 + 6 • So, x = 5 or - 5 So, x = 2 + 6 or 2 - 6

  29. Completing the Square (Example 1) If x2 + bx is a binomial then by adding b2 which is the square of half 2 the coefficient of x, a perfect square trinomial results: x2 + bx + b2 = x + b2 2 2 Solving a quadratic equation with ‘completing the square’ method. Example: Step1: Isolate the Binomial x2 - 6x + 2 = 0 -2 -2 Step 2: Find ½ the coefficient of x (-3 ) x2 - 6x = -2 and square it (9) & add to both sides. x2 - 6x + 9 = -2 + 9 (x – 3)2 = 7 x – 3 = + 7 x = (3 + 7 ) or (3 - 7 ) Note: If the coefficient of x2 is not 1 you must divide by the coefficient of x2 before completing the square. ex: 3x2 – 2x –4 = 0 (Must divide by 3 before taking ½ coefficient of x) Step 3: Apply square root method

  30. (Completing the Square – Example 2) Step 1: Check the coefficient of the x2 term. If 1 goto step 2 If not 1, divide both sides by the coefficient of the x2 term. Step 2: Calculate the value of : (b/2)2 [In this example: (2/2)2 = (1)2 = 1] Step 3: Isolate the binomial by grouping the x2 and x term together, then add (b/2)2 to both sides of he equation. Step 4: Factor & apply square root method 2x2 +4x – 1 = 0 2x2 +4x – 1 = 0 (x + 1) (x + 1) = 3/2 2 2 2 2 (x + 1)2 = 3/2 x2 +2x – 1/2 = 0 (x2 +2x ) = ½ √(x + 1)2 = √3/2 (x2 +2x + 1 ) = 1/2 + 1 x + 1 = +/- √6/2 x = √6/2 – 1 or - √6/2 - 1

  31. Quadratic Formula General Form of Quadratic Equation: ax2 + bx + c = 0 Quadratic Formula: x = -b + b2 – 4ac discriminant: b2 – 4ac 2a if 0, one real solution if >0, two unequal real solutions if <0, imaginary solutions Solving a quadratic equation with the ‘Quadratic Formula’ 2x2–6x + 1= 0 a = ______ b = ______ c = _______ x = - (-6) + (-6)2 – 4(2)(1) 2(2) = 6 + 36 –8 4 = 6 + 28 = 6 + 27 = 2 (3 + 7 ) = (3 + 7 ) 4 4 4 2 2 -6 1

  32. Solving Higher Degree Equations x3 = 4x x3 - 4x = 0 x (x2 – 4) = 0 x (x – 2)(x + 2) = 0 x = 0 x – 2 = 0 x + 2 = 0 x = 2 x = -2 2x3 + 2x2 - 12x = 0 2x (x2 + x – 6) = 0 2x (x + 3) (x – 2) = 0 2x = 0 or x + 3 = 0 or x – 2 = 0 x = 0 or x = -3 or x = 2

  33. Solving By Grouping x3 – 5x2 – x + 5 = 0 (x3 – 5x2) + (-x + 5) = 0 x2 (x – 5) – 1 (x – 5) = 0 (x – 5)(x2 – 1) = 0 (x – 5)(x – 1) (x + 1) = 0 x – 5 = 0 or x - 1 = 0 or x + 1 = 0 x = 5 or x = 1 or x = -1

  34. Rational Expressions Rational Expression – an expression in which a polynomial is divided by another nonzero polynomial. • Examples of rational expressions • x 2 • x 2x – 5 x – 5 • Domain = {x | x  0} Domain = {x | x  5/2} Domain = {x | x  5}

  35. Multiplication and Division of Rational Expressions A • C = A 9x = 3 B • C B 3x2 x 5y – 10 = 5 (y – 2) = 5 = 1 10y - 20 10 (y – 2) 10 2 2z2 – 3z – 9 = (2z + 3) (z – 3) = 2z + 3 z2 + 2z – 15 (z + 5) (z – 3) z + 5 A2 – B2 = (A + B)(A – B) = (A – B) A + B (A + B)

  36. Negation/Multiplying by –1 -y – 2 4y + 8 y + 2 4y + 8 -y - 2 -4y - 8 - = OR

  37. Examples x3 – x x + 1 x – 1 x x2 – 25 x2 –10x + 25 x2 + 5x + 4 2x2 + 8x •  x2 – 25 2x2 + 8x x2 + 5x + 4 x2 –10x + 25 • (x3 – x) (x + 1) x(x – 1) = = (x + 5) (x – 5) • 2x(x + 4) (x + 4)(x + 1) • (x – 5) (x – 5) = = x (x2 – 1)(x + 1) x(x – 1) 2x (x + 5) (x + 1)(x – 5) = x (x + 1) (x – 1)(x + 1) x(x – 1) = (x + 1)(x + 1) = (x + 1)2 =

  38. (x + 1) (x –7) (x + 1) (x – 1) 1 x2 + x - 6 x – 2 3 • 1 (x + 3) (x – 2) x – 2 3 (x – 7) (x – 1) • (x + 3) 3 Check Your Understanding Simplify: x2 –6x –7 x2 -1 Simplify: 1 3 x - 2 x2 + x - 6 

  39. Addition of Rational Expressions Adding rational expressions is like adding fractions With LIKE denominators: 1 + 2 = 3 8 8 8 x + 3x - 1 = 4x - 1 x + 2 x + 2 x + 2 x + 2 (2 + x) (2 + x) 3x2 + 4x - 4 3x2 + 4x -4 (3x2 + 4x – 4) (3x -2)(x + 2) = = = 1 (3x – 2)

  40. Adding with UN-Like Denominators • + 2 • x2 – 9 x + 3 • 1 + 2 • (x + 3)(x – 3) (x + 3) • 1 + 2 (x – 3) • (x + 3)(x – 3) (x + 3)(x – 3) • 1 + 2(x – 3) 1 + 2x – 6 2x - 5 • (x + 3) (x – 3) (x + 3) (x – 3) (x + 3) (x – 3) • + 1 • 8 • (3) (2) + 1 • 8 • + 1 • 8 • 7 • 8 = =

  41. x – 1 (x + 1)(x –1) = = 1 (x + 1) Subtraction of Rational Expressions To subtract rational expressions: Step 1: Get a Common Denominator Step 2: Combine Fractions DISTRIBUTING the ‘negative sign’ BE CAREFUL!! 2x - x + 1 x2 – 1 x2 - 1 2x – (x + 1) x2 -1 2x – x - 1 x2 -1 = =

  42. b b-1 2(b – 2) b-2 - b -b+1 2(b – 2) b-2 + b 2(b – 2) 2(-b+1) 2(b – 2) + -1 2 -1(b – 2) 2(b – 2) b –2b+2 2(b – 2) -b + 2 2(b – 2) = = = Check Your Understanding Simplify: b b-1 2b - 4 b-2 -

  43. x + 2 3x - 1 x x + 4 Complex Fractions A complex fraction is a rational expression that contains fractions in its numerator, denominator, or both. Examples: 1 5 4 7 x x2 – 16 1 x - 4 1 x 2 x2 + 3 x 1 x2 - 7/20

  44. (2x – 1) (x - 2) (x + 1) Rational Equations 3x = 3 x + 1 = 3 6 = x 2x – 1 x – 2 x - 2 x + 1 3x = 3(2x – 1) 3x = 6x – 3 -3x = -3 x = 1 x + 1 = 3 x = 2 6 = x (x + 1) 6 = x2 + x x2 + x – 6 = 0 (x + 3 ) (x - 2 ) = 0 x = -3 or x = 2 Careful! – What do You notice about the answer?

  45. (12x) 6 (x + 1) -3(x – 1) = 4x 6x + 6 –3x + 3 = 4x 3x + 9 = 4x -3x -3x 9 = x = Rational Equations Cont… To solve a rational equation: Step 1: Factor all polynomials Step 2: Find the common denominator Step 3: Multiply all terms by the common denominator Step 4: Solve x + 1 - x – 1 = 1 2x 4x 3

  46. (4x2) (x + 2)(x – 2) 4x + 4 = 3x2 3x2 - 4x - 4 = 0 (3x + 2) (x – 2) = 0 3x + 2 = 0 or x – 2 = 0 3x = -2 or x = 2 x = -2/3 or x = 2 3(x + 2) + 5(x – 2) = 12 3x + 6 + 5x – 10 = 12 8x – 4 = 12 + 4 + 4 8x = 16 x = 2 Other Rational Equation Examples 3 + 5 = 12 x – 2 x + 2 x2 - 4 1 + 1 = 3 x x2 4 3 + 5 = 12 x – 2 x + 2 (x + 2) (x – 2)

  47. Solve for p: • = 1 + 1 • F p q 1 x - 1 2(x – 3) x(x – 2) 3 x(x – 1)(x + 1) Check Your Understanding Simplify: x 1 x2 – 1 x2 – 1 1 3 x – 2 x 1 1 2 x(x – 1) x2 – 1 x(x + 1) Solve 6 1 x 2 3 2 2x – 1 x + 1 2 3 x x – 1 x + 2 x2 + x - 2 4 + - = 1 = 5 - + = + - -1/4 Try this one:

  48. Radical Equations Continued… Example1: x + 26 – 11x = 4 26 – 11x = 4 - x (26 – 11x)2 = (4 – x)2 26 – 11x = (4-x) (4-x) 26 - 11x = 16 –4x –4x +x2 26 –11x = 16 –8x + x2 -26 +11x -26 +11x 0 = x2 + 3x -10 0 = (x - 2) (x + 5) x – 2 = 0 or x + 5 = 0 x = 2 x = -5 Example 2: X2= 64 Example 3:

  49. Inequality Set & Interval Notation Set Builder Notation {1,5,6} { }  {6} {x | x > -4} {x | x < 2} {x | -2 < x < 7} x such that x such that x is less x such that x is greater x is greater than –4 than or equal to 2 than –2 and less than or equal to 7 Interval (-4, ) (-, 2] (-2, 7] Notation Graph -4 0 2 7 -2 Question: How would you write the set of all real numbers? (-, ) or R

  50. Inequality Example StatementReason 7x + 15 > 13x + 51 [Given] -6x + 15 > 51 [-13x] -6x > 36 [-15] x < 6 [Divide by –6, so must ‘flip’ the inequality sign Set Notation: {x | x < 6} Interval Notation: (-, 6] Graph: 6

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