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Trinomials are 3 sets of numbers that are not alike in any ways. In order to solve trinomials, you need to know your factors and distributive property! x 2 13 x 42 c 12 c 35 To get x 2 , you have to multiply x times x (x ) (x ) Since the last term is , the signs will be the same as the middle term. (x ) (x ) Now, the factors of 42 are : 1 and 42, 2 and 21, 3 and 14, 6 and 7. The right facor needs to equal to 13 when it' s added, it also needs to be equal to 42 when it' s multiply! (x 6) (x 7) Use the distributi ve property! x(x) x(7) 6(x) 6(7) y 2 13 y 30 x 2 7 x 6 x 42 combine like terms d 7 d 10 x 2 13 x 42 Try these examples: x 12 x 27 2 2 2 Rose Felisme Finding the Equation of a Line Given Two Points X Y -3 Step 1: Find the slope of the line containing the points. 6 m= y2-y1 Slope formula x2-x1 = -4-(-1)(x1,y1)=(-3,-1) and (x2,y2)=(6,-4) 6-(-3) = -3 or-1 Simplify. 9 3 Step 2: Use the slope and one of the two points to find the yintercept. y = mx+b Slope-intercept form -4 = -1(6) + b Replace m with -1,x with 6, and y with -4. 3 3 -4 = -2 + b Multiply. -2 = b Add 2 to each side. Step 3: Write the slope-intercept form using m=-1 and b=-2 y = mx+b Slope-intercept form 3 y = -1x-2 Replace m with -1, and b with -2. 3 3 -1 -4 Problems for you to try: A. (2,3) (6,4) B. (2,4) (2,1) C. (-1,12) (4,-8) D. (36,15) (22,10) Description: If you know two points on a line, first find the slope. Then follow these steps… •Use the slope and one of the two points to find the yintercept •Write the slopeintercept form using y=mx+b •Check your answer by graphing the line. It should pass through the two points. How to do it: The table shows the coordinates of two points on the graph of a linear function. Alexis Clark Simplifying square Roots using perfect square square Roots is a quantity of which a given quantity is the square. Perfect squares are a rational number that is equal to the square of another rational number. EXAMPLE 81 2 y 9 y The first thing you need to know while simplifying square roots using perfect squares is if the number your dealing with is a perfect square. In this case our number is 81 and its a perfect square because 9 times itself gives you 81. The second thing you need to do is find the square roots of y2 which is y because y times itself is equal to y2 9y Lastly you just put it all together and the answer is 9y Practice Problems 2 25 ba 36 b 22 169 ba 4 121 x Kennie Rebecca Solving inequalities using all operations An inequality can be use when we don’t know what an expression is equal to… instead of an equal sigh we an use this symbol < ≥ > ≤ Four practice problems 1)5x-8<12 2)4-2x≤2x-4 3)-13m>-26 4)14g>56 For example:12x-4<8 12x<12(add 4 to each side of the inequality) X<1 ( divide both sides of the inequality by 12) One important rule you should always remember if you multiplying or dividing both sides of an inequality by a negative number reverse the direction of the inequality sigh Tamarre Cynthia Jabouin 2+4y > -6 2+4y < 6 -4 -3 -2 -1 0 1 2 3 Nia Rogers Description: A way to graph a equation in slopeintercept form, the equation is y=mx+b: m is the slope and b is the y-intercept and y and x is the points. Slope is the steepness of the line and intercept is where the line connects and intersects with the y For Example: axis. y 5x4 1 y x ( 5 ) Slope Intercept Form 5 1 1 y x ( 5 ) slope isand y intercept is 5 5 5 First Graph intercept and the then slo t For y9x3 Y intercept 4 Y intercept 3 Rise Run 5 1 Example 1 y x 5 5 1 y x 3 9 y 11 x 4 : Fall-9 Run1 Gary Chen 5/31/12 Factoring a trinomial with leading coefficient other than 1 Description- This is one way to factor a polynomial. In this case there is a trinomial and also a leading coefficient other that 1. Factoring this is the inverse of the distributive property that would result in two binomials Example 10d2 + 17d - 20 1 -multiply the numbers of the two outside terms and find factors that would equal the middle term 10 x -20= -200 2- Find two fractions who’s sum add up to 17 -8 and 25 are factor of -200 that adds up to 17 3 substitute the middle term with the fractions 10d2-8d + 25d-20 4 is to split the problem so it is separated by the middle operation (10d2-8d )+ (25d-20) GCF= 5 GCF=2d Find the GCF of both sides of the equation 3-Distribute the GCF to each of the terms to each equation using division instead of multiplying 2d(5d-4) + 5(5d-4) Now add the outside terms and multiply it to the inside term (The terms inside the parentheses should be the same in each part of the problem. (2d-5) (5d-4) Answer Practice Problems By Jamari Robinson 2 10x +21x-10 12q2+34q-28 8z2+20z-48 12y2-4y-5 Factoring a trinomial with leading coefficient other than 1 Description- This is one way to factor a polynomial. In this case there is a trinomial and also a leading coefficient other that 1. Factoring this is the inverse of the distributive property that would result in two binomials Example 10d2 + 17d - 20 1 -multiply the numbers of the two outside terms and find factors that would equal the middle term 10 x -20= -200 2- Find two fractions who’s sum add up to 17 -8 and 25 are factor of -200 that adds up to 17 3 substitute the middle term with the fractions 10d2-8d + 25d-20 4 is to split the problem so it is separated by the middle operation (10d2-8d )+ (25d-20) GCF= 5 GCF=2d Find the GCF of both sides of the equation 3-Distribute the GCF to each of the terms to each equation using division instead of multiplying 2d(5d-4) + 5(5d-4) Now add the outside terms and multiply it to the inside term (The terms inside the parentheses should be the same in each part of the problem. (2d-5) (5d-4) Answer Practice Problems By Jamari Robinson 2 10x +21x-10 12q2+34q-28 8z2+20z-48 12y2-4y-5 Adding and Subtracting Polynomials Example : Problem 1 2 2 2 (30x 10 x 8 ) ( 10 x 20 x 4 ) 40 x 30 x 12 2 2 2 2 Guid : 30x 10 x 40 x / 10x 20 30 8 4 12 4 3 x / 1 When adding polynomials you must find all like terms for each variable, exponent and co-efficient and you must add them with each one that is the same term. Example : Example : 2 2 ( 60 x 12 x 3 ) ( 30 x 2 x 1 ) 2 2 ( 20 x 10 x 2 ) ( 10 x 20 x 1 ) Subtraction for polynomials wouldn’t be too different you would just find the variables, the exponents and the coefficient and instead of adding you would subtract. -William Higgins Example : 2 2 ( 20 x 10 x 2 ) ( 10 x 5 x 1 ) Example : 2 2 (50x 20 x 5 ) ( 14 x 10 x 5 ) Solving equations with multi step. I am going to talk about how to solve multi step equation as you can this equation below is a multi step equation. It contain coefficient and one variable and distribute to clear the properties 4(50 3x) 80 20 200 12x 80 20 Distribute the parentheses 200 12x 100 Simplify the equation 12x 100 Subtract both sides 12x 100 Divide both sides 12 12 you divide them x 8 . 3 3 After you get the answer. Practice problems 3(3x5)49 3x519 x62 x4010347 2(910 x)3040 Guervens Charles Finding the equation of a line given two points Description To be able to solve this concept, there is two things that you need to know. First thing you need to know is how to find the slope of a line using two points. The second thing you need to know is how to find the yintercept of a line using two points. 4 practice problems (-2,0) (8,4) (-3,0) (3,3) (2,4) (4,8) (8,16) (16, 32) Example • (-1,0) (1,4) • Find the slope Slope= • Slope= 40 4 2 1 1 2 y2 y1 x2 x1 • Find the y-intercept • You have to solve for b (y-intercept) • Let (1,4) be x and y y mx b 4 2(1) b Replace the letters by their values 4 2 b Multiply the slope and the x value - 2 - 2 Minus two on both sides 2 b The answer is 2. • The equation is y=2x+2 Rubens Lacouture Multiplying A Binomial By A Binomial Kasie Okafor ( x 3 )( x 2 ) x x x 2 3 x 3 2 Add The Result: x x x 2 3 x 3 2 2 x 2 x 3 x 6 2 x 5 x 6 ( m 4 )( m 5 ) ( y 2 )( y 8 ) ( x 5 )( x 7 ) ( x 3 )( x 4 ) The Answer Solving Inequalities using all operations Description: An Inequalities is the condition of being unequal lack of equality disparity. 3 x 9 6 x 12 3x - 3x 9 3x - 12 12 12 21 3x 4 Practice Problems: 3 3 2x+ 6 > 4x- 16 7 x 9x -5 < 45x +12 8x-9>7x+12 x 7 8x-5<23x+13 First you -3x to both sides , then you have 9>3x-12 you have to +12 to both sides. Then you have 21\3 >3x\3 you cross 3x\3 out and 7> x . Your answer x>7. Jennifer Jean-Louis Multiplying polynomials A trinomial has three terms and a binomial has two terms, but they are all polynomials. To multiply polynomials you have to multiply each term to every other term. An example of this is: 3 ( x 2 8 )( 4 x 8 2 ) Step 1: To do this problem, you have to take the first term which is x cubed, and multiply that by all the other terms in the trinomial next to it. (4x+8+2) You have to continue to do this with each term in the first trinomial. Once you do this, you will get this for the answer. You get this by combining all the like terms. 33 34 3 3 x 4 x x 8 x 2 4 x 8 x 2 x Step 2: 2 4 x 2 8 2 2 8 x 16 4 4 Practice Problems: 2 2 ( 7 x x 3 )( 7 8 )( 3 x 1 )( x 2 ) Step 3: 8 4 x 8 8 8 2 32 x 64 16 5 x ( x 3 ) 4 x ( x 3 8 ) 2 Answer: 3 4 10 x 4 x 40 x 100 Mykala Jordan 3x4y 25 Multiply by2. 2x3y 6 Multiply by-3. ()-6x9y-18 6x8y-50 17y -68 Add theequations. 17y -68 Divide each side by17. 17 17 y-4 Simplify. Now Substitute -4foryineither equation ofind t out the value of x. 2x-3y6 Second Equation 2x-3(-4) 6 y-4 2x126 Simplify. 2x12-126-12 Substract 12from each side. 2x-6 Simplify. 2x -6 2 2 x -3 1 a .)5x3y6 2x5y10 3c.) 6x -2y 10 Divide each side by2and Simplify. 3x -7y -19 The solution is(-3,-4) 2b.) 6a 2b 2 4a 3b 8 4d.) 9p q13 3p 2q -4 By: Mark Britt Distributive property to simplify and solve expressions. one way is th pr at 12x(4y oblem th 2x) eth you an 12x an take d multiply by 4y an th den th 12 an multiply e take d it by 2x you 2 distr d th pr ibute e 12x oblem. 4y 48xy an 12x 2x d 24x th e 12 x (4 y 2 x ) an is 48xy swer 24x . ( 12 x 4 y ) ( 12 x 2 x ) 48 xy 24 x 2 expression A mathematical equation which can contain numbers, operators the four operation and variable (like x, y) to represent equation or a operation. Like 12x(4y+2x) 5 x 5 15 Solving multi-step equations with 5 5 Practice problems operations 7m17 60 5 x 10 / Description, 5 /5 Practice problems 5 x 5 15 examples x 2 all 1. To solve the 7 x 4 32 2.Then divide 5 to 5x, problem above, you and 10, and you will subtract 5 from both get x=2. The variable sides. is 2. 3 b 4 13 • Multi-step equations are equations that takes more than 1 steps to solve that specific equation. Such as some examples below. They just basically requires more work. Here are some examples 8 3 r 7 2 a 6 4 Jackson C. Ngo Finding the equation of a line given two points Use the following coordina s, practi to find th eq 1. (6,1) (8,3) 2. (6, 1) (9,0) 3. (5,10) (10,15) 4. (7,21) (8,24 Tofind theequationof a linegivenby twopoints,youuse twocoodinates tofind theslope, andfind they - interceptbyseeingwherethelinecrossesin thegraph, or pluggingthecoordinate to theequationyoufound. Tofind theslope,youusetheequation, 2 1 2 1 y y x x in someequationswherea gridis available, youcanusetherise/runmethodtoalsofind theslope, they is therise thexis therun. Wewillbeusingthecoordinate s, (1,2)and(4,5). 2 2 (4,5) 1 1 Coordinate (4,5)wouldbe y andx whileCoordinate (1,2)wouldbe y andx (1,2) 5- 2 3 Yourslopewouldbe3 because 3 dividedby1equals3. 2 -1 1 Theformof theequation ewwantis y mx b (bbeingthey intercept, m beingtheslope) Soyourequation ould w be y 3x b. Tofind they - intercept, you would substitute a coordinate , into thecurrentequation. So,y 3x b usingthecoordinate (4,5)it becomes , 5 34 b , 5 12 b 12- 5 7 , therefor youry intercept is 7. y 3x -7 By: Junior Tatis Multiplying a Polynomials by a Monomials Polynomials are just two or more monomials added together. When an degree is asked for a polynomial its usually asking for the highest exponent for a variable. Examples: (p3)(p24p2) p(p2)p(4p)p(2)3(p2) In order to do all these you would have had to had known distributive property p35p2p3p2 p33p3p2 2 (x 2 )( x 2 x 1 ) 2 2 x (x ) x ( 2 x ) x ( 1 ) 2 (x ) 2 ( 2 x ) 2 ( 1 ) Try these, real fun! (2x 2)(x2 2x 1) (4x 1)(x2 4x) (4x 2)(x 4x 2) (3x 3)(x5 9) 3 2 2 x 2 x x 2 x 4 x 2 3 x 3 x 2 Practicing will prepare you for success on the Math Finals! Tatyana Adams Solving Systems of equations using 4 Practice substitution problems Example y 3x x 2 y 21 2x 7 y 3 x 1 4 y x 2 y 21 x 2 ( 3 x ) 21 x 3 y 12 x y 8 x 6 x 21 7x 21 x 3 y 3x Simplify, Combine the like terms Divide each side 7 Use y 3x to find the value of y y 3x y 3(-3) or - 9 The solution is (-3,-9) Check the solution by graphing or put the solution in the problem to see if it makes sense 6x 2 y 4 y 3x 2 a b 1 5a 3b 1 Description: Use the substitution method to eliminate one of the variables in your equation. When you find the answer to one of the variables plug it in the equation to find the other variable. Nedcar Faugas is an algebra property which is used to multiply a single term and two or more terms inside a set of parentheses. Take a look at the problem below. example2 3 ( x x 1 ) 2 2 3 ( x x 1 ) 3 ( x ) 3 ( x ) 3 ( 1 ) 2 3 x 3 x 3 Practice these four- 1( x 2 x) 3( x 2 x 4 ) 7( x x) 4(3 x 2 x 2) Solving Inequalities Using All Operations 10 n 63 4 n 27 A multi step equa 10n 63 4n 27 First step isto get variable the by itself 4n 4n Subtract he like variables t 6 n 63 27 Then you add the terms EXAMPLES 63 63 6 n 90 Simplify and divide by six 66 n 15 Solving Inequalities using all operations is when you use all the steps and operations you learned to solve inequalities. You need to know how to add and subtract like terms. Also you would need to know how to simplify equations. You will also need to graph them on a number line. 2n20 4n32 6x12 2x 64 9v 183v21 10g 505g 25 Karan Richards Multiplying a Polynomial by a Monomial 2 2 2 x ( 3 x 7 x 10 ) Dis Us ve Pr 2x2(3x27x10 ) 2 2x2(3x2) ( 2x2)( 7x) (-2x )( 10 ) - Distribute the Polynomial to each ofthe Monomials in the parenthesi s. 3 6x4( 14x ) ( 20 x2) - After you distribute the Polynomial to the Monomials make sure that your multiplica tions are correct. Make sure that your exponents are correct and that your negatives and positives are in the right places. 6x4 14 x3 20 x2 - Multiply he new Polynomial t by the Monomials in the parenthesi s ( Use distributi ve property !!!!!!!)And make sure you have the correct exponents and that your negatives and positives are in the right place and Ifyou did your multiplica tions correct, you should have your answer :) Try some for yourself :D 2 2 23 2 2 2 5 a ( 4 a 2 a 7) 5y( 2y 7 y ) 6x ( 5 3 x 11 x ) 4 x ( 5 x 1 x 7 y ) To solve this equation you need to use the distributive property. Distribute the Polynomial to all the Monomials in the parenthesis. Yanick Cardoso DIVIDING MONOMIALS •First take the 12 and divide it by the three. 12 divided by 3 is 4. •Then were going to take the a’s from up top, and at the bottom and see how many pairs we can cross out. We crossed out 3 pairs so at that point. Now it’s a raised to the 3rd power. So now were going to do the same to the next variable which is C. 9 12 12a c 3 6 4a c aaaaaaaaa aaa Practice Problems 5 4 3 4 7 10 s 5s3 25 g 6 5g 10 13 30y f 6 6 5y f 10 cccccccc cc cccccc We crossed out 6 pairs so that is going to be C raised to the 6th power. At the end your answer should look like this. 3 6 4ac Jhlyik Lezama Multiplying a polynomial by a monomial We are going to do “multiplying a polynomial by a monomial”. Where we multiply whatever is in the parenthesis with the outside variable. Our first step is to multiply both Here is our variables inside the parenthesis with example the outside parenthesis 8(2x -6) After multiplying it would give us our 8(2x) 8(-6) final answer of 16x – 48 16x - 48 With that you are done and free to try some practice problems below. 4 (6 x 8) 9 (5 x 3) 8(4 x 8) 6 (6 x 4 ) Vu Nguyen. Multiplying a polynomial by a monomial. 2 x ( 3 x 7 x 10 ) 22 First thing is the distributive property. 2 2 2 2 2 x ( 3 x ) ( 2 x )( 7 x ) ( 2 x ) 1 ) The next thing is to multiply 6 x ( 14 x ) ( 2 x ) 4 Examples: 3y(5y2) 2x(4a4 3ax 6x2) t(5t 9)2t 3 2 After you multiplied you have to simply 4 3 2 6 x 14 x 2 x 4xy (5x2 12 xy7y2) Brittany Odom Example I Solve for x Solving System of Equation Using Substitution When you Solving System of equation using substitution you have to solve for “y” and “x”. Because solving system of equation you know you have to Practice Problem I find two solution. 3 y 2 x 11 y 4x 2 6 x 1y 4 6 x 1( 4 x 2) 4 6x 4x 2 4 2x 2 4 2x 2 x 1 Substitute y=4x-2 into the 1y Distribute Multiplication Subtract 6x-4x Add 2 both side Divide both side by 2 Solve for Y y 4x - 2 Sub the X answer from the previous Problem y 4(1) - 2 Multiplication y 4-2 Subtract 4-2 to find Y y2 Answer is (1,2) y 9 2x Practice Problem II y 3x 2 8x 2 y 4 Fredens Altine Solving Systems Of Equations Using Substitution Steps to Solve Systems by Substitution • Solve one of the equations for y •Substitute the expression for y into the other equation. •Solve for x. •Substitute the value of x into either of the original equations to find the value of y •Write the solution as a coordinate pair. x - 2y = 14 x + 3y = 9 a. First, be sure that the variables are "lined up" under one another. In this problem, they are already "lined up". b. Decide which variable ("x" or "y") will be easier to eliminate. In order to eliminate a variable, the numbers in front of them (the coefficients) must be the same or negatives of one another. Looks like "x" is the easier variable to eliminate in this problem since the x's already have the same coefficients. c. Now, in this problem we need to subtract to eliminate the "x" variable. Subtract ALL of the sets of lined up terms. (Remember: when you subtract signed numbers, you change the signs and follow the rules for adding signed numbers.) d. Solve this simple equation. Try these .. e. Plug "y = -1" into either of the ORIGINAL equations to get the value for "x". •4x + 3y = -1 5x + 4y = 1 f. Check: substitute x = 12 and y = -1 into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE! •4x - y = 10 2x = 12 - 3y •x - 2y = 14 x + 3y = 9 P = 2 + 2Q P = 10 – 6Q x - 2y = 14 x + 3y = 9 x - 2y = 14 x + 3y = 9 x - 2y = 14 -x - 3y = - 9 - 5y = 5 -5y = 5 y = -1 x - 2y = 14 x - 2(-1) = 14 x + 2 = 14 x = 12 x - 2y = 14 12 - 2(-1) = 14 12 + 2 = 14 14 = 14 (check!) x + 3y = 9 12 + 3(-1) = 9 12 - 3 = 9 9 = 9 (check!) SHAY WEBSTER Adding and Subtracting polynomials When you are adding/subtracting polynomials, you have to add/subtract like terms. When you are done adding like terms you have to put them in order from largest to smallest, answers with exponents always go first. EXAMPLES! 1. (2x+3y)+(4x+9y) {add the x’s first then the y’s} 6x+12y Practice problems for you to try…. • (5+4n+2m)+(-6m-8) •(5a+9b)-(2a+4b) •(5f+g-2)+(-2f+3) •(11m-7n)-(2m+6n) 2. (6s+5t)+(4t+8s) {add the s’s first then the t’s} 14s+9t Jonique Tabb Solving Two Step Equations One goal in solving an equation is to have only variables on one side of the equal sign and numbers on the other side of the equal sign. The other goal is to have the number in front of the variable equal to one. The variable does not always have to be x. These equations can make use of any letter as a variable. 1. 3x+10=100 3x+10=100 2. 7x+10=52 Were basically going to undo the probably by doing the opposite. So first I’m going to subtract 10 from both sides. That’s going to leave me with 3x=90. -10 -10 3x=90 _ _ 3 3 X=30 Then were going to divide both sides by 3. 3x divided by 3 leaves you with x, and 90 divided by three leaves you with 30. Your problem should be finished off with x equaling something which is 30 so x=30. 7x+10=52 -10 -10 7x=42 _ _ 7 Subtract 10 from both sides. Divide both sides by 7 7 X=6 There’s your answer Practice Problems 1. 3x+5=14 2. 2x - 3=-9 3. 3x-2=10 4. 3x+5=14 Cookie Bourne Factoring Trinomials Factoring a trinomials means finding two binomials that when multiplied together It makes a trinomial. This is kind of like the opposite of multiplying two binomials. 1x²-10n+25 1x²+15n+14 1x²-8n-48 1x²+p-20 EXAMPLE 1x² + 5x – 36= (x + 9) * (x – 4) SOLUTION The problem asks me to factor the trinomial into two binomials. STEP 1 List out all the factor of the number with no variable. STEP 2 Add/Subtract the factors of that number and see if it adds up to the middle number. STEP 3 After that turn them into two binomials. Jimmy Lai Graphing the Solution of an inequality on a number line EXAMPLE: Graphing inequalities on number line represent the solution to inequalities . It aids in visualizing the answer. Graphing inequalities is simple once you learn the few simple steps to solve a problem. PRACTICE PROBLEMS Graph: x < 4 Solution: The problem asks you to graph all numbers that are less than 4. STEP1: Draw an open circle on the number 4. (Don’t draw a CLOSED circle because it does have the _ under the symbol. STEP2: Draw a line going left, because x is less then 4. HERES THE SOLUTION X<-9 x > 24 | 2x + 3 | < 6 5 >x Parmanand Jodhan Multiplying A Binomial By A Binomial My project is to multiplying a binomial by a binomial. When you multiply a binomial by a binomial you have to multiply every number in the problem by each other. Practice problems (6 – 2x) (10 – 7x) (12x + 4) (3x + 7) (8x – 9) (14x – 13) (9x+ 24) (5x + 1) First you do 3x times 5x and get 15x. Next you do 2 times 4 and you get 8.after that its 2 times 5x the answer is 10x.lastly you do 3x times 4 and it equals 12x. Alisha Cooper