CHAPTER+4

OPERATIONS WITH MONOMIALS, POLYNOMIALS

//Adding and Subtracting: -Bryan// //When simplifying polynomials get all like terms together. Simplify: 3x- //x2 //-9+2x^2+-5x // Answer= x2-2x-7
 * //To add two or more polynomials, write their sum and then simplify by combining similar terms.//
 * //To subtract one polynomial from another, add the opposite of each term of the polynomial you're subtracting.//
 * So get the x squares together= x2
 * Then get the x's together= -2x
 * Then get the numbers together= -7

 //Add: x^2y^2-8xy^4+2x^4y+5x^3y^3+2xy^4

//      //<span style="FONT-FAMILY: 'Arial','sans-serif'">Subtract: 3x^2+4x-5 from 2x^2-9x+7 // <span style="FONT-FAMILY: 'Arial','sans-serif'"> <span style="FONT-FAMILY: 'Arial','sans-serif'"> <span style="FONT-FAMILY: 'Arial','sans-serif'">__-3x^2+4x-5__ <span style="COLOR: #00b050; FONT-FAMILY: 'Arial','sans-serif'">-x^2+-13x+2 <span style="FONT-FAMILY: 'Arial','sans-serif'"> <span style="FONT-FAMILY: 'Arial','sans-serif'"> <span style="FONT-FAMILY: 'Arial','sans-serif'"> <span style="COLOR: red; FONT-FAMILY: 'Arial','sans-serif'">
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">First, you only can add like terms. So add -8xy^4+2xy^4 because x is to the first power and y is to the fourth power and you get -6xy^4. If one of the x’s was to a different power you could not add them. Like-8x^3y^4+ 3xy^4.
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">Now you cannot add anymore because none of them are like terms
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">So the answer is -6xy^4+2x^4y+5x^3y^3+2x^4y
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">2x^2-9x+7
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">So I put 2x^2-9x+7 over 3x^2+4x-5 because it said to subtract that from that.
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">Also make sure you subtract like terms.
 * **//<span style="COLOR: red; FONT-FAMILY: 'Arial','sans-serif'">Definition //** || **//<span style="COLOR: red; FONT-FAMILY: 'Arial','sans-serif'">New Examples //** ||
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">Constant || <span style="FONT-FAMILY: 'Arial','sans-serif'">-2, 3/5,0 ||
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">Monomial || <span style="FONT-FAMILY: 'Arial','sans-serif'">-7, u, 1/3m^2, -s^2t^3, ^xy^3 ||
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">Coefficient || <span style="FONT-FAMILY: 'Arial','sans-serif'">4n^2 4___x__ 1 ___ -m^3n^2__ -1_ ||
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">Degree || <span style="FONT-FAMILY: 'Arial','sans-serif'">10xy^3: degree of x 1 and y 3 ||
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">Degree || <span style="FONT-FAMILY: 'Arial','sans-serif'">9f^4g^3h__8__ ||
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">Similar || <span style="FONT-FAMILY: 'Arial','sans-serif'">-s^2t^3 and 2s^2t^3 are similar: 6xy^3 and 6x^3y are Not! ||
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">Polynomial || <span style="FONT-FAMILY: 'Arial','sans-serif'">Ex: x^2+ (-4)x+5 or x^2=4x+5 ||
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">Simplified Polynomial || <span style="FONT-FAMILY: 'Arial','sans-serif'">2x^3-4x+2-5x+3x^3-7=5x^3-9x-5 ||
 * <span style="FONT-FAMILY: 'Arial','sans-serif'">Degree || <span style="FONT-FAMILY: 'Arial','sans-serif'">Ex: above has degree 3 (simplify FIRST!) ||

EXPONENT RULES-Bryce
 * < a^m*a^n ||< (a)^(m+n) ||
 * < (ab)^n ||< (a)^(n)(a)^(m) ||
 * < (a^m)^n ||< (a)^(mn) ||
 * < (a)^0 ||< 1 ||


 * Examples :**

2^5=32 2^2*2^3=32 (2^2)^=64 4x^4*x^3=2x^7 (x^m)^n*x^p= x^m^n+^p

http://www.purplemath.com/modules/exponent.htm

FACTORING 7 METHODS 2x^2+6x 2x (x + 6) 169y^4-625x^8 (13y^2 - 25x^4) (13y^2 + 25x^4) 4x^2 + 8xy + 7x + 14xy 4x (x + 2y) + 7(x + 2y) (x + 2y) (4x + 7)
 * Greatest Common Monomial Factor
 * DOTS
 * Grouping
 * Cubes (Difference, Sum)

Difference of Two Cubes (x^3 -8) (x-2) (x^2 + 2x + 4) Sum of Two Cubes (8y^3 + 27) (2y + 3) (4y^2 - 6y + 9)

x^2 + 3x + 2 (x + 2) (x + 1) 2x^2 + 9x + 4 (2x + 1) (x + 4) [<span class="wiki_link_new">(a^2 + 2ab + b^2) ] (9x^2 + 24xy + 16y^2) (3x + 4y)^2
 * Short Trinomial
 * Long Trinomial
 * Perfect Square Trinomial


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Pascal's Triangle-Bryce Pascal's Triangle is named after Blaise Pascal because he invented it.



Use Pascal's Triangle to expand (x+4)^7

First you would go to the 7 row on the pascals triangle and find the numbers that are in that row then you would put then in order like so- **//__1__//** (x) ^7(4)^0+**__//7//__** (x)^6(4)^1 and so on. Second you would put the (x) then (4), After that you start with the highest exponents so x to the 7 power and 4 to the 0 power then you go down one from the 7 and up one from the 0. x goes to the 6 and 4 goes to the 1 and so on. Then u start to solve when you get x is to the 0 and 4 is to the 7

1(x)^7(4)^0+7(x)^6(4)^1+21(x)^5(4)^2+35(x)^4(4)^3+35(x)^3(4)^4+21(x)^2(4)^5+7(x)^1(4)^6+1(x)^0(4)^7 (x)^7+28x^6+336x^5+2240(x)^4+8960(x)^3+21504(x)^2+28672x+16384

http://ptri1.tripod.com/

SOLVING QUADRATIC EQUATIONS BY FACTORING- Bryan

ax^2+bx+c=0 is called a Quadratic Equation. When we solve a quadratic equation, the equation is always set equal to zero. We use the ZERO PRODUCT PROPERTY which states that if ab=0 if and only if a=0 orb=0 (At least one of its factors must equal 0) The solutions are called ROOTS or ZEROS Note: The degree of the polynomial determines the number of solutions. ( Quadratic: 2, Cubis: 3, Quartic: 4, Quintic: 5) However, it should be noted that the roots may not all be within the set of real numbers! A graphing calculator will be helpful in determining how many of the roots are real.

Examples:

Solve: x^2-2x=8 A. Set equal to 0 x^2 -2x-8=0 B. Factor (x-4) (x+2) C. Set each factor = 0 D. State solution { 4,-2 }

(a+3)(a-3)=40 a^2-49=0 (a-7)(a+7) {-7,7}

x^2+25=10x x^2-10x+25 (x-5)^2 {5 d.r.} There are two fives so it is a double root

1/6x^2+1/2x-2/3=0 multiply everything by 6 because you need a common deminator x^2 +3x-4=0 (x+4) (x-1) { -4,1 } http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-827999-2&chapter=6&lesson=3&headerFile=10&state=pa

Find the Zeros of: t.r.= triple root d.r.= double root

f(t)=(t-2)^3 (t-3)^2 0= {2 t.r., 3 d.r.}

f(x)= (x-1)^4 -4(x-1)^3 (x-1)^3 (x-1-4) x-5 { 1 t.r., 5}

SIGN GRAPHS
 * Sign graphs ae used to solve quadratic inequalities
 * The first thing you must do is factor the inequality as you would an equation
 * You then find the solutions
 * Make one more number line than you have solutions
 * Draw a vertical line down through all of your lines and place a plus or minus in each section (The + or - is determined whether that section would give you a positive or negative answer)
 * Your solution is then the - signs if you inequality was a less than or the + if it was a greater than

WORD PROBLEMS


 * 1) 1 Two numbers differ by 8 and have a product of 84. Find the number.

First, make the number "x" Then, the second by "x + 8”, because it says it differs by 8. After that, multiply x(x+8) and set it equal to 84 You then simplify to x2+8x-84=0 Then, you factor to (x-6) (x+14) =0 Your solutions are {-14, 6} Your two possible sets of numbers are {6,14} and {-6, -14}


 * 1) 2 The product of three consecutive integers is 21 more than the cube of the smallest integer. Find the integers.

First Integer- x; Second Integer- (x+1); Third Integer- (x+2)

x(x+1)(x+2)=x^3+21 x^3+3x^2+2x+2=x^3+21 3x^2+2x-21=0 (3x-7) (x+3)=0

x={(7/3), -3}

(-3,-2,-1)

A batter hits a baseball directly up with a speed of 96ft/s. A. How long is the ball in the air before being caught by the catcher? B. How high does the ball go?
 * 1) 3 Use the formula h=rt-16t^2

First, replace 96 in the formula for r because r is the rate/speed 0=96t-16t^2 0=16t(6-t) {0,6} So for letter A the answer is 6 seconds

For B you have to divide the time by 2 which is 3 Plug 3 in the formula for t which is time h=96(3)16(3)^2 h=288-144 h=144 ft So letter B is 144 feet

<span style="MARGIN-TOP: 29px; Z-INDEX: 251662336; MARGIN-LEFT: 458px; WIDTH: 140px; POSITION: absolute; HEIGHT: 114px"> #4 A right triangle has a hypotenuse of 25. The sum of the two legs is 31. Find the two legs.

x2 + (31-x)2 = 252 x2 + (x2-62x+961) = 625 2x2 – 62x + 336 = 0 2 (x2 -31x + 168) = 0 2 (x-7) (x-24) = 0 x= {7, 24}

Your only solution is (7, 24).


 * 1) 5. The length of a rectangle is one meter less then twice the width. If the area is 55 square meters, find the perimeter.

x=width first you would multiply the width and length together and set it equal to "55" -- x(2x-1)=55 then solve 2x^2-x=55 2x^2-x-55=0 2x^2-11x+10x-55=0 x(2x-11)+5(2x-11)=0 (x+5)(2x-11) (-5, 11/2)- ( -5 cant work cause you cant have a negative side) (11/2)


 * //5.5m * 10m//**