Algebra+tutorial Factoring(2)

Algebra 1 -tutorial ---factoring and solving equations Factoring is a simple procedure...You collect similar or like terms and fish out common 'factors'.The rest are put within brackets.After doing a few examples, you will feel comfortable...not much theory, but some imagination on your part to 'factor out' the various terms. Grouping Let us take a simple example: y = 3 X - 12 Consider the two terms, 3x and 12. We can write 12 = 3 x4 What is common to both the terms?: number 3 Let us 'factor out' 3 from both the terms. We get: y = 3X - 3x4 = 3 (x - 4 ) 2. Let us see another example; z = ax+ay + bx + by Look at the first two terms: a is common to both; likewise, 'b' is common to third and fourth terms. Therefore z = a ( x + y) + b ( x + y ) now look at the equation again. (x + y) is common to both the terms: z = ( a + b ) ( x + y) Expanding the terms -- FOIL method. Often we have expressions which are products to two terms: y=( a +b )( c+d ) Then expanding means we get four terms: First: multiply the first quantity in both brackets: a.c Outer: the two outer quantities: a.d Inner: the two quantities inner,close to each other: bc Last: multiply the second quantity in each bracket: bd. Therefore y = ac + ad + bc + bd. This is the FOIL method which you learn after some practice: Here are some examples : 1 . (x + 2 ) ( x + 3) = x^2 + 3x + 2x +6 = x^2 + 5x + 6 2.

(3x -2) ( 2x - 1) F = 3x.2x=6x^2 O= 3x (-1) = -3x I = (-2)(2x) = -4x L = (-2)(-1) = 2 So, we get (3x-2) (2x-1) = 6x^2 -3x -4x + 2 = 6x^2 - 7x + 2 3.

(2x + 3)(5x-4) = 10 x ^2 - 8x +15x - 3(-4) = 10X^2 + 7x - 12. Do it Yourself! 4.

( 5x - 1) (2x + 3) =

5.

( X + 3) (x + 4) =

6.

( a + 5) ( a - 3) =

Factoring by Guessing?! Many times we factor expressions by guessing some numbers.Consider this example: x.x + 5x +6 = 0

You have to guess two numbers, say , m and n so that m+n = 5 [the middle number] m.n = 6 [the third number] Guess what? You know the numbers. m = 3 n=2 Roght? Now you simply write: (x+m)(x+n) =0 ( x + 3) (x+ 2) = 0 Check this with expanding it---use foil method: (x + 3)(x+2)= x.x + 2x + 3x + 6 = x.x + 5x + 6 You can solve this equation easily: x+3 =0 or x = -3 and x+2 = 0 or x = -2 The two solutions are: x = -3 , x = -2 1.Let us see some more examples to get the method of guessing under our mental grip. x.x + 6x + 9 = 0 Now guess two numbers, m & n; m = 3 & n = 3 Right! [The expression is called TRINOMIAL because it has three (tri) terms!] (x+3 ) ( x + 3) =0 or (x + 3 ) ^ 2 = 0 Solving we get: x+3 =0 or x = -3 {Only one solution here.] 2. Factor and solve :

x.x + x - 6 = 0

Again guess two numbers: One number can be negative because m.n= -6 Guess what? m+n = 1 Think for a moment: m=3 n= -2 -----Got it right! ( x + 3) ( x -2)=0 Check with FOIL method. Solving: x= -3 and x=2 are the solutions. 3. Factor and solve: x.x - 6x + 8 = 0 Here m+n = -6 m.n=8 Since m.n is positive, both the numbers must have the same sign. Let m = -4 n= -2 Right?! The problem is done now! (x-4)(x -2) = 0 or x=4 , and x = 2 Study the three examples once again.You find that m and n can be both positive numbers or one of them negative or both negative....Keep this in mind..the guessing becomes easier... Now for some "Do It Yourself" problems: Factor and solve: 4. x.x + 7x + 12 = 0 5. x.x + 7x + 10 =0 6. x.x + 7x + 6 = 0 7. x.x - 7x + 12 =0 8. x.x - 7x +10 = 0 9. x.x - x -30 = 0 10. x.x + x - 12 = 0 Some interesting word problems 1 Sarah's age is 6, his brother is elder to him.The sum of their ages is equal to 15. Find the age of her brother. Let her brother's age be x years. The sum of their ages = Sum = 6 + X = 15 6 + X = 15 6 -6 + x = 15 - 6 X=9 Her brother's age is 9. 2. The sum of two consecutive odd numbers is 24 .Find the numbers. Let the number be X. The consecutive odd number is x+2. x + ( x + 2) = 24 x + x + 2 = 24 2 x + 2 = 24

2x + 2 -2 = 24 - 2 2x = 22 x = 22/2 = 11 x+2 = 13 The consecutive odd numbers are: 11 and 13. 3.The length of a rectangular park is 9 yards more than the width .The area of the park is 36 square yards. Find the length and width of the park. The area of the park is length x width. Let the width be x. The length of the park = x + 9. Area = length x width = x (x + 9) = 36 x.x + 9 x = 36

method]

x.x + 9 x - 36 = 0 (x + 12) ( x-3) = 0

[ Check this step with FOIL

x+ 12 = 0 or x =-12 Now the width cannot be negative! so we reject this solution. Take x - 3 =0 or x = 3 ...therefore the width of the park is 3 yards and length is x+9 or 12 yards. 4 The length of a rectangular garden is 5 more than 3 times the width.The area is 42 square feet.Find the length. Start with width. Let the width be W. Then length is : L = 3 W + 5 Now the area A = L x W = ( 3 W + 5 ).W = 3WW + 5W = 42 3WW +5W - 42 = 0 We have to solve the above equation and find W. 3 ww + 14w - 9w -42 = 0 ( 3w +14) ( w - 3)=0 The solution 3w+ 14=0 means that w=-14/3 ----Since width cannot be negative,we reject this solution. But w - 3 =0 is Okay w=3 Therefore width is 3 feet, length is : L = 3w+ 5 = 3 x 3 + 5=14 feet. Check: area = length x width = 14 x 3 = 42 square feet. 5 A business problem is taken up here: A company making calculators sells x pieces per week.The price of each calculator is p.As price increases, the sales drop, a common experience in any business. John ,the manager, finds that x is related to p as follows: x= 1700 - 100 p If John wants a weekly revenue of $7000, what should be the price? Revenue is the money received by selling goods. Revenue R = price x number sold R = p.x = p ( 1700 - 100 p) = 7000 Therefore 1700 p - 100 p.p = 7000 Dividing by 100, we get 17p - p.p = 70 or 17 p - p.p - 70 = 0 or p.p - 17p + 70 = 0 Let us solve this equation: p.p - 10p - 7p +70 = 0 (p - 10)(p - 7) = 0 Therefore p- 10 = 0 or p=10 is one solution. The other solution is : p- 7 = 0 or p=7 Therefore John can sell at prices either $10 or $ 7. Why two solutions? Think for a moment. John can sell at the lower price;it will bring in more customers! 6 A Physics problem here: A rocket is fired vertically with a pseed of 176 feet per second.Its height h at any time t after firing is given by the equation: h = -16 t.t + 176 t At what time, the rocket will be at 480 feet height? Let us first set up the equation: 480 = - 16t.t + 176 t

16 t.t -176 t + 480 = 0 Divide by 16 : t.t - 11 t + 30 = 0 Now factor the equation be guesing two numbers: ( t - 6 )( t - 5) = 0 Solve now! t = 6 or t = 5 Well, the height of the rocket reaches 480 feet at 5 seconds and at 6 seconds...Why two solutions!....think smart, dummy! the rocket goes up and comes down! ----------------------------------------------------------------------------------------------------------To sum up: -- Use distributive property or FOIL method to expand the expression. --- Use grouping to factor --- Use number guessing for trinomials to factor and solve. There are faster methods you will use later on. ---With practice comes speed. -------------------------------------------------------------------------------------------------------