31 – Basic Algebra Review Rules for Order of Operations To make sure an expression is always evaluated in the same way by different people, the Order of Operations convention was definedMnemonic: “Please Excuse My Dear Aunt Sally”ParenthesesExponentsMultiply/DivideAdd/SubtractAlways: Evaluate & Eliminate the innermost grouping first
113 – Solving Systems of Equations Using the Substitution Method
123 – Solving Systems of Equations Using the Elimination (Addition) Method
133 – Solving Systems of Equations Solution to 3 Equations Adding (A) and (C) will eliminate y (A) 2x – y + 3z = 6 (C) 2x + y + z = (D) 4x +4z = first new equation in 2 variablesAdding (B) and 5·(C) will also eliminate y (B) 3x – 5y +4z = 7 5·(C) 10x + 5y + 5z = (E) 13x + 9z = second new equation in 2 variablesSolve (D) and (E) like a system of two equations (next page)Use Substitution or Addition
14Solution to (D) 4x + 4z = 4 Continued (E) 13x + 9z = -3 Well use substitution of x from (D) into (E) to find z (D) 4x +4z = (D1) x = 1 - z move 4z to the other side, divide by 4Substitute x from (D1) into (E) (E) x + 9z = (1 – z) + 9z = – 13z + 9z = use distribution, then simplify z = z = 4Substitute z into (D) or (E) or (D1) to find x (D) 4x + 4(4) = x + 16 = x = x = -3Substitute x and z into (A) or (B) or (C) to find y (C) 2(-3) + y + (4) = y + 4 = y = 6 Solution is (-3, 6, 4)
154 – Inequalities Intersections, Unions & Compound Inequalities Set DiagramsIntersections of SetsConjunctions of Sentences andUnions of SetsDisjunctions of Sentences orInterval NotationDomains
164 – Inequalities Expressing Domains With Interval Notation
174 – Inequalities Using the Absolute Value Principle |x + 1| = 2 x + 1 = 2 or x + 1 = -2|2y – 6| = 0 2y – 6 = 0|5x – 3| = -2 no solution
184 – Inequalities When an equation has 2 absolute values?
195 – Polynomials & Factoring Subtracting Polynomials To subtract polynomials, add the opposite of the second polynomial.(7x3 + 2x + 4) – (5x3 – 4) add the opposite! (7x + 2x + 4) + (-5x3 + 4)Use either horizontal or vertical addition.Sometimes the problem is posed as subtraction: x2 + 5x make it addition x2 + 5x (x2 + 2x) _ of the opposite x2 – 2x__ x +6
205 – Polynomials & Factoring Multiplying Two Polynomials To multiply a polynomial by a polynomial, we use the distributive property repeatedly.Horizontal Method:(2a + b)(3a – 2b) = 2a(3a – 2b) + b(3a – 2b)= 6a2 – 4ab + 3ab – 2b2 = 6a2 –ab – 2b2Vertical Method: 3x2 + 2x – 54x + 26x2 + 4x – 1012x3 + 8x2 – 20x____12x3 + 14x2 – 16x – 10
215 – Polynomials & Factoring FOIL: Used to Multiply Two Binomials
245 – Polynomials & Factoring Using a Factor Table - Trial & Error Let’s use x2 + 13x as an exampleFactors must both be sums: (x + ?)(x + ?)Pairs=c= Sum=b=131,2,3,4, ok quit!x2 + 13x + 36 = (x + 4)(x + 9)
25The ac Grouping Method: ax2 + bx + c Split bx into 2 Terms: Use a Table based on a·c Let’s use 3x2 – 10x – 8 as an exampleac = 3(-8) = -24 One factor is positive, the other negative and larger.Pairs=ac= Sum=b=-101,2, quit!3,4,3x2 – 10x – 8 =3x2 + 2x – 12x – 8 = split the middlex(3x + 2) – 4(3x + 2) = do grouping =(3x + 2)(x – 4)
265 – Polynomials & Factoring Factoring Perfect Square Trinomials? x2 + 8x + 16 = (x + 4)2(x) (4) (x)(4) = 8x yes, it matchest2 – 12t + 4 = not a PST(t) (-2) (t)(-2) = -4t no, it’s not -12t25 + y2 + 10y = (y + 5)2y2 + 10y descending order(y) (5) (y)(5) = 10y yes, it matches3x2 – 15x + 27 = not a PST3(x2 – 5x + 9) remove common factor(x) (-3) (x)(-3) = -6x no, it’s not -5xPST Tests:1. Descending Order2. Common Factors3. 1st and 3rd Terms (A)2 and (B)24. Middle Term 2AB or -2AB
275 – Polynomials & Factoring Difference of Squares Binomials Remember that the middle term disappears? (A + B)(A – B) = A2 - B2It’s easy factoring when you find binomials of this pattern A2 – B2 = (A + B)(A – B)Examples:x2 – 9 =(x)2 – (3)2 =(x + 3)(x – 3)4t2 – 49 =(2t)2 – (7)2 =(2t + 7)(2t – 7)a2 – 25b2 = two variables squared(a)2 – (5b)2 =(a + 5b)(a – 5b)18 – 2y4 = constant 1st, variable square 2nd2 [ (3)2 – (y2)2 ] =2(3 + y2)(3 – y2)
326 – Rational Expressions & Functions Multiplying Fractions (Use parentheses for clarity)Factor expressions, then cancel like factors
336 – Rational Expressions & Functions Dividing Fractions Change Divide to Multiply by Reciprocal, follow multiply procedure
346 - Finding the LCD (must be done before adding or subtracting 2 or more RE’s) 1. Factor each denominator completely into primes.2. List all factors of each denominator. (use powers when duplicate factors exist)3. The LCD is the product of each factor to its highest power.28z3 = (22) (7)(z3)21z = (3)(7)(z)LCD=(22)(3)(7)(z3)a2 – = (a + 5)(a – 5)a + 7a + 10 = (a + 5) (a + 2)LCD = (a + 5)(a – 5)(a + 2)
356 - Adding or subtracting rational expressions with unlike denominators 1. Find the LCD.2. Express each rational expression with a denominator that is the LCD.3. Add (or subtract) the resulting rational expressions.4. Simplify the result if possible.
376 - Simplifying Complex Rational Ex’s Method 2: Multiplying by Reciprocal Making the top and bottom into single expressions, then multiplying by reciprocal.
386 – Rational Expressions & Functions Rational Equations: False Solutions Solve a Rational Equation by Multiplying BOTH SIDES by the LCDWarning: Clearing an equation may add a False SolutionA False Solution is one that causes a divide by zero situation in the original equationBefore even starting to solve a rational equation, we need to identify values to be excludedWhat values need to be excluded for these?t ≠ a ≠ ± x ≠ 0
396 – Rational Expressions & Functions Clearing & Solving a Rational Equation What gets excluded?x ≠ 0What’s the LCD?15xWhat’s the solution?
406 – Rational Expressions & Functions Dividing a Polynomial by a Polynomial Use the long-division process
417 – Radical Expressions & Functions Examples to Simplify
427 – Radical Expressions & Functions Practice Express using rational exponents:Simplify using rational exponents:
43Rational Expressions Where the Numerator is greater than 1 Using Exponent Arithmetic, it’s a little easier
447 – Radical Expressions & Functions The Product Rule for Radicals
457 – Radical Expressions & Functions The Quotient Rule for Radicals
467 – Radical Expressions & Functions A Radical Expression is Simplified When: Each factor in the radicand is to a power less than the index of the radicalThe radicand contains no fractions or negative numbersNo radicals appear in the denominator of a fraction
477 – Radical Expressions & Functions Definitions A Radical Equation must have at least one radicand containing a variableThe Power Rule:If we raise two equal quantities to the same power, the results are also two equal quantitiesIf x = y then xn = ynWarning: These are NOT equivalent Equations!
487 – Radical Expressions & Functions Equations Containing One Radical To eliminate the radical, raise both sides to the index of the radical
49Equations Containing Two Radicals Make sure radicals are on opposite sidesSometimes you need to repeat the process
507 – Radical Expressions & Functions i, The Basis of the Complex Number System
528 – Quadratic Functions The Square Root Principle Solve by factoringx2 – 16 = (x+4)(x-4)=0 x=4,-4Then by the square root propertyx2 – 16 = x2 = x=4,-4
538 – Quadratic Functions Using Completing the Square to Solve an Equation
548 – Quadratic Functions Introducing … The Quadratic Formula! The Quadratic Formula is used to find solutions to any quadratic equationThe formula was derived using completing the square and the square root property.
558 – Quadratic Functions Solving Quadratic Form Equations The variable u is often used to replace squares of variables or expressions