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Expanding and Factorizing

Expanding and Factorizing. By: The New Girls Ingrid Orellana and Marcela García. SECTION 1: Expanding Brackets. Expanding expressions in Brackets. What’s expanding expressions in brackets?

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Expanding and Factorizing

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  1. Expanding and Factorizing By: The New Girls Ingrid Orellana and Marcela García

  2. SECTION 1:Expanding Brackets

  3. Expanding expressions in Brackets • What’s expanding expressions in brackets? • Well, this is the term used for the expressions that contain brackets and can be re-written without those “brackets” It Looks confusing but it is really easy. Lets look at the next examples: • 4(x + y) = 4x + 4y a(2 - b + 4c) = 2a - ab + 4ac

  4. 4(x + y) = 4x + 4y a(2 - b + 4c) = 2a - ab + 4ac In the first example the number which is outside the brackets (4 in the first case, and a in the second case) is multiplied by the numbers inside the brackets. This is the reason that each term on the right side of the first example starts with a 4, and why each term on the second example starts with an a How to expand expressions in Brackets

  5. Now! Try to do next exercises anLet's Test Your Knowledge ...

  6. a.5(2x+2) = b.6(4x+2y) = c.4y(10x+5) = a.10x+10 b.20x+12y c.40xy+ 20y If you got themALL right, then you can move into the Next Stage!

  7. Now, Lets Look At A More complicated example: 2m (mnp + 3m2) = 2m2np + 6m3 That was harder than the first one right? Well, to start with the2and themoutside the brackets are multiplied by each number inside thebrackets. As you can see the first term inside the brackets already has anm, so when this is multiplied by the m outside the brackets, it becomesm2. At the same time, multiplyingthe m2in the second term bymturns it intom3. The 2 is also multiplied by each term inside the brackets. But, BECAREFUL! Because in the(+3m2)has a3at the start, so the 2must be multiplied by it, giving an answer of 6(2*3)‏

  8. BUT… what happens when the number isnegativeor the sign used isminus? Let’s look at the next example

  9. -5p2q2(10pq - 7p3q) = -50p2q2 + 35p5q2 This looks kind of hard... but actually it’s easy. To start with, by looking at the example above you will notice that the numbers inside the bracket are being multiplied by anegativenumber (-5 rather than just 5). This in fact affects the signs, but how? First,-5p2q2*10pqbecomes -50p2q2, and -5p2q2 *-7p3qbecomes+ 35p5q2(because negative multiplied by a negative number gives a positive number)SO be careful with this! – This particular thing has caught many people many times!

  10. Consider the following expression in brackets: (x + y)(m + n) We multiply the first term in the first bracket by the first term in the second bracket to givexm We the multiply the first term in the first bracket by the second term in the second bracket, which will givexn Expanding quadratic expressions

  11. Now, you have to multiply the second term in the first bracket by the first term in the second bracket to giveym To end up, we multiply the second term in the first bracket by the second term in the second bracket to giveyn To remember all of the steps just remember that each term must be multiplied by the terms found in the second bracket or to make simpler, just remember to make a happy face when making the lines…Like this That was easy Right?

  12. Factorizing Expressions

  13. To factorize an expression means to put it into brackets. It is basically the opposite of expanding brackets

  14. If you have an expression like this one: 10a3b-20a2 +5a3bc-15ab2 And you are asked to factorize it by finding the common factor, this is what you shall do..

  15. Step 1: • What is the factor that each of these have in common? • (5) • Step 2: • What else do they have in common? • The variable a2 • When you have found out this, you will put it outside the brackets like this: 5a2() • Step 3: • Then, you find out what the common factor has been multiplied by. When you have done this, put the multiples inside the brackets • Your answer should be as follows: • 5a(a2b-4+a2bc-3b2)

  16. Test Yourself(Answers at the end) • Factorize these expressions by common factor: • 8a3 -4a • 12c2 +2ac • 5y3 +15y2 • 49x3-21x2

  17. Method 2: Perfect Trinomial • A Perfect Trinomial will always have three terms. • Example: • a2+6a+9 • To factorize this expression you need to do the following…

  18. Fully Factorize the expressiona2+6a+9 • First, you find the square root of the first term, then of the third term. (a and 3) • These are the numbers that arer going to go on your brackets (a+3) (a+3) • To simplify this expression you just need to write it out as (a+3)2 • What about the second term? • To check if your expression is right, divide teh second expression by 2. 6a/2=3a. 3a is 3 of the first term, times a of the second term. • Therefore, your answer is correct!

  19. Continuation • Some Trinomials you can think of this way: • x2 - 12x + 32 • Ask yourself: • What two numbers, that when they are multiplied together give 32 and when they are added they give 12? • The answer is 4 and 8 because 8 times 4 is 32 and 8+4 is 12. This satisfy the conditions of the expression and therefore are correct. • Your final answer should be: • (x-8) (x+4)

  20. Try some more out… • 16x2- 8xy + y2 • a2+2ab+b2 • x2+12x+36 • x2+7x+12 • a2+9a+20

  21. Method 3: Difference of Squares • If there are only two terms in the expression, then the answer will be positive and negative for the same number. • Example: • x2+16 • (x+4) (x-4)

  22. More examples…Try them out! • x2+25 • 4x2+49 • 6x2+9 • 8x2+81

  23. A Final Tip! • After you have factorized the expression CHECK that the answer you have, gives you the expression that you have just factorized.

  24. Answers • Common Factor: • a)4a(2a-1) • b)2c(6c+a) • c)5y2(y+3) • d)7x2(7x-3) • Perfect Trinomials • a) 4x-y)2 • b)(a+b)(a-b) • c)(x+6)2 • d)(x+4)(x+3) • e)(a+5)(a+4) • Difference of Squares • a)(x+5) (x-5) • b)(2x+7)(2x-7) • c)(2x+3)(3x-3) • d)(4x+9)(2x-9)

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