Transcript 6x 3 - Quia
Chapter 4 Polynomials Exponents 20= 1 21=2= 2 22=2*2= 4 23=2*2*2= 8 24=2*2*2*2= 16 25=2*2*2*2*2= 32 28=256 Special Cases 1. Anything to the zero power is 1 x0 = 1 2. Second power is also called “squared” 3. Third power is also called “cubed” Monomial A monomial is an expression that is a number (constant), variable or the product of a number and one or more variables. x2 x is the base, 2 is the power Examples 1. 2. 3. 4. 5. 6. 5 X 5x y3 5x3 5x3y2 Combining Like terms Like Terms are monomials with the same variables all raised to the same power. You can only add and subtract like terms Like Terms Not like Terms 2x + 3x = 5x 2a + 3b x2 + 3x2 = 4x2 2x2 + 2x3 x2y3 + 2x2y3 = 3x2y3 x2y4 + 2x3y6 Multiplying Monomials am * an = am+n x3 * x 5 = x8 x2 * x 3 = x5 2x2 * 3x5 = 6x7 (3n2)(-4n4) = -12n6 1. 2. 3. Multiply the numbers (constants) Take each variable in alphabetical order When the bases are the same add the powers. Powers of Monomials 1. 2. 3. Power of a Power (am)n = amn (x4)5 = x20 (x3)2 = x6 [(-a)2]3 = (a2)3 = a6 1. 2. 3. 4. Power of a Product (ab)m = ambm (2x)3 = 8x3 (3x2y)2 =9x4y2 (-2a)5 = -32a5 (8x3y4z5)2 = 64x6y8z10 Polynomials A polynomial is the sum of monomials. Polynomials are usually written with their variables in alphabetic order, with highest powers first. Examples 1. 2x - 9 2. 3x2 + 2x - 3 3. 4x3y + 3x2y2 - 2xy3 +4 4. 2a + 3b2 – 4c +5d3 Multiplying Polynomials by Monomials Distribute to each term, then combine like terms. 1. x(x+3) = x2 + 3x 2. 3y(2y+4) = 6y2 + 12y 3. -2x(4x2 - 3x + 5) = -8x3 + 6x2 - 10x 4. 5xy2(3x2 - 4xy + y2) = 15x3y2 - 20x2y3 + 5xy4 Multiplying Polynomials 1. 2. 3. 4. 5. (x+3)(x+2) = x2 (x+3)(x+2)=x2 + 2x (x+3)(x+2)= x2 + 2x + 3x (x+3)(x+2)= x2 + 2x + 3x + 6 (x+3)(x+2)= x2 + 5x + 6 First Outer Inner Last Simplify Multiplying Polynomials (3x-2)(2x2-5x-4) = 6x3 2. (3x-2)(2x2-5x-4) = 6x3-15x2 3. (3x-2)(2x2-5x-4) = 6x3-15x2-12x 4. (3x-2)(2x2-5x-4) = 6x3-15x2-12x-4x2 5. (3x-2)(2x2-5x-4) = 6x3-15x2-12x-4x2 +10x 6. (3x-2)(2x2-5x-4) = 6x3-15x2-12x-4x2+10x+8 6x3-15x2-12x-4x2+10x+8 = 6x3-19x2-12x+10x+8 6x3-19x2-12x+10x+8 = 6x3-19x2-2x+8 6x3-19x2-2x+8 1. Transforming Formulas a = ½ bx solve for b a * 2 = 1/2bx * 2 2a = bx 2a / x = bx / x 2a/x = b b = 2a/x c = ax – b solve for x c + b = ax – b + b c + b = ax (c + b) / a = ax / a (c + b) / a = x x = (c + b) / a A Problem Solving Plan 1. 2. 3. 4. 5. Read the problem carefully, making a sketch may help. Choose variables for the facts given. Reread the problem and write an equation. Solve the equation Check your results. Rate, Time and Distance D=R*T Bicyclists Brent and Jane started at noon from points 60 km apart and rode toward each other, meeting at 1:30pm Brent’s speed was 4 km/h greater than Jane’s speed. Find their speeds. It took them 1.5 hours to meet. If Jane is going ‘j’ fast then Brent must be going ‘j +4’ They traveled 60 m. 1.5(j+4) + 1.5j = 60 Solving Jane = 18 Brent = 22 Area Problems A=L*W A rectangle is three times as The original area minus long as it is wide. If it 36 is equal to the new length and width are area. both decreased by 2, L = 3w then its area is decreased w(3w) – 36 = (w-2)(3w-2) by 36. Find its original Solving we find w = 5, dimensions. so l = 15. Problems with no solution Not all word problems have solutions. Here are some reason for this: 1. Not enough information is given. 2. The given fact lead to an unrealistic result. 3. The given facts are contradictory.