Algebra Review

The Language of Algebra Miss Jones is planning on going to Paris. Every week, she puts a certain amount of money into a fund for her trip. Her grandparents gave her $75 to add to the fund. After 12 weeks she has a total of $255. How much does she put into the fund each week? x = the amount Miss Jones saves each week 12x = the amount she saved so far (12 weeks) 255 = 12x + 75 -75 - 75 whatever you do to one side 180 = 12x on an equation you MUST 15 = x do to the other Add this

Combining with geometry Many problems are related through geometric formulas. Don’t forget that these are on the formula sheet located in the test booklet. EX. Mr. Carter has 500 feet of fencing and wants to enclose a rectangular plot along the river to protect his vegetable garden. What is the largest plot he can enclose. (1) DRAW A PICTURE and LABEL IT River x y (2) THINK ABOUT WHAT YOU KNOW fencing goes around something so we are considering perimeter but we want to maximize the area (4) Substitute and Solve the equations A = LW A = x y A = (x)(500 – 2x) A = 500x - 2x2 this is an inverted parabola so there will be a maximum Either use the calculator and trace the max value Or complete the square for vertex form remember that 15625 is impacted by the -2 A = -2(x2 -250x ) A = -2(x-125)2 A = -2(x – 125) + 31250 Vertex is at (125, 31250) So an x of 125 gives the largest area enclosed 31250 square ft. (3) Set up equations to relate the variables modifying if necessary P = L + 2W Enter values you know 500 = y + 2x 500 - 2x = y -31250 +15625 -31250

Adding & Subtracting Polynomials Add & Subtract LIKE terms (5x2 + 3) – (2x2 + 4x – 7) (5x2 + 3) + (-2x2 – 4x + 7) 3x2 – 4x + 10

Multiplying Polynomials Multiply every term in the first polynomial by every term in the second polynomial, then combine like terms (5x2 + 3)(2x2 + 4x – 7) (multiply all terms by 5x2) (then multiply all terms by 3) 10x4 + 20x3 – 35x2+ 6x2 + 12x – 21 Now add like terms 10x4 + 20x3 – 29x2 + 12x – 21

Evaluating equations (extension) x2 – 2x + 5 = y find y when x = 1 (1)2 – 2(1) + 5 = y 1 – 2 + 5 = y 4 = y Find x when y is doubled x2 - 2x + 5 = 8 x2 – 2x - 3 = 0 (x – 3)(x + 1) = 0 x-3= 0 x+1=0 x = 3 x = -1 You could use the quadratic equation!

Solving Equations Whatever you do to one side, you MUST do to the other! 3(x-4) – 6 = 2x + 8 3x – 12 – 6 = 2x + 8 distribute 3x – 18 = 2x + 8 combine like terms + 18 +18 move #s to one side 3x = 2x + 26 - 2x -2x variables to the other x = 26 divide if necessary

Degree of a Polynomial The degree of the highest term Found by adding all the exponents in a term together Example The degree of 3x2y3 – 4xy2 degree of each term 5 3 5th degree equation

Terminology Binomial – two terms in a polynomial x + 1 Trinomial – three terms in a polynomial x2 + 2x + 3 Linear equation an equation with degree =1 y = kx Quadratic equation an equation with degree = 2 y = ax2 + bx + c