Expansion of Terms. f (x) = x 2 6x + 9 = (x 3) 2 = 0. x 3 = 0

Transcription

1 Expansion of Terms So, let s say we have a factorized equation. Wait, what s a factorized equation? A factorized equation is an equation which has been simplified into brackets (or otherwise) to make analyzing it a bit easier. For example, by factorizing some quadratic equations, we don t need to use the quadratic formula to find values that will make the equation equal zero. Example 1: Using the knowledge that the function f (x) = x2 6x + 9 = (x 3) 2, calculate the values of x where f(x) = 0. Method 1: The Quadratic Equation f (x) = x 2 6x + 9 = 0 a = 1,b = 6,c = 9 x = b ± b2 4ac 2a x = ( 6) ± ( 6)2 4(1)(9) 2(1) x = 6 ± x = 6 ± 0 2 x = 3 Method 2: Using the Factorized Equation f (x) = x 2 6x + 9 = (x 3) 2 = 0 ( x 3) 2 = 0 x 3 = 0 x = 3 We can clearly see here that factorizing equations is clearly beneficial, sometimes. The equation is easier to write up, we can analyze points quicker and more efficiently. Question: What happens when we need to analyze specific terms in an equation? Well, we end up with problems. What do we do when we need to add something like (x 4)2 into the equation? We can t just combine the two equations, that d be ugly. Instead, what we do is expand the two equations. No, not like balloons, but there s a few techniques we can use that will stop our expanded equations from ballooning out.

2 Technique One: F.O.I.L. The first technique, F.O.I.L, is used exclusively for equations with two brackets with two terms each in them. For example, a factorized equation such as (x 4)(x + 3). The idea is simple. To expand the two brackets, we need to multiply each term on the left hand bracket with each term on the right hand bracket. Now, if we just did that, we might get the answer, or lost along the way. F.O.I.L. is a way to make things significantly easier. It s an acronym for expanding, and it goes something like this: First Outer Inner Last The idea is, that by following this order, you ll cover all of the terms, and not multiply something twice (or even the wrong numbers). Let s apply this: Example 2: Expand the function f (x) = (x 4)(x + 3). f (x) = (x 4)(x + 3) F = x x O = x 3 I = ( 4) x ( )( 3) L = 4 f (x) = F + O + I + L f (x) = x 2 + 3x 4x 12 f (x) = x 2 x 12 Identify each expanded term in F.O.I.L. Swap f(x) for F+O+I+L. Simplify. ONLY use F.O.I.L. when you re expanding two brackets. Keep in mind that things like (x 2)(x 3)3 do not contain two brackets, but actually contain more. If you have a bit of trouble with using F.O.I.L., you can always try thinking of using a different acronym, or trying your own methods. Just be careful, if you end up making an acronym that makes a lot of careless mistakes, it may not be the best idea to use it.

3 Technique Two: Expansion Rules So you ve found an equation that you couldn t expand using F.O.I.L. It s not the end of the world, don t worry. There are actually a few techniques to expand equations. The next one we will be learning is fairly straightforward. They re called the expansion rules. Basically, there s a few rules we can follow to expand equations easily. They re listed just below. Quadratic Equations (a + b) 2 = a 2 + 2ab + b 2 (a b) 2 = a 2 2ab + b 2 (a + b)(a b) = a 2 b 2 Cubic Equations (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 (a b) 3 = a 3 3a 2 b + 3ab 2 b 3 (a b)(a 2 + ab + b 2 ) = a 3 b 3 Now, these formulas are straightforward to use, but only apply to quadratic and cubic functions. Now, let s apply these. Example 3: Expand f (x) = (x 4)2. f (x) = (x 4) 2 Find the rule that applies to the question. (a b) 2 = a 2 2ab + b 2 Define the terms of a and b. a = x b = 4 Expand f (x) = x 2 2(4)(x) + (4) 2 f (x) = x 2 8x +16 Now, we can apply this to any number of two termed expressions, provided that they re in this form. However, for two termed expressions that are taken to the power of four, or five, or even more, we re unable to use either technique. So, we use technique three. It s called binomial expansion.

4 Technique Three: Binomial Expansion The binomial expansion goes a little like this: (a + b) n = a n + n C 1 (a n 1 )(b 1 ) + n C 2 (a n 2 )(b 2 ) + n C n 2 (a 2 )(b n 2 ) + n C n 1 (a 1 )(b n 1 ) + b n where n! n C x = x! ( n x)! So, when we are unable to use the first two techniques, and we need to expand things with to a power > 4, we use the binomial expansion. Let s try an example. Example 4: Expand f (x) = (x 4)5 f (x) = (x 4) 5 Utilize the Binomial Expansion formula. f (x) = x C 1 (x 4 )( 4) + 5 C 2 (x 3 )( 4) C 3 (x 2 )( 4) C 4 (x)( 4) 4 + ( 4) 5 f (x) = x 5 + (5)( 4)(x 4 ) + (10)(16)(x 3 ) + (10)( 64)(x 2 ) + (5)(256)(x) + ( 1024) f (x) = x 5 20x x 3 640x x 1024 Note: We can only apply this to factorized equations where we have two terms in the brackets. We can t expand something like (a + b + c)n with this, that uses a different method altogether (which we don t need to know). But I can t use any of these! Got an equation like g(x) = (4x + 3)(2x2 + 6x + 3)? Ouch, you must be having a bad day. Not to worry. Even though we can t use any of these techniques as is, we can modify the F.O.I.L. method to suit our needs. Just remember -> Each term on the left hand side needs to be multiplied with a term on the right hand side, just once. Then the terms need to be collected. Example 5: Expand g(x) = (4x + 3)(2x2 + 6x + 3). g(x) = (4x + 3)(2x 2 + 6x + 3) g(x) = (4x)(2x 2 + 6x + 3) + 3(2x 2 + 6x + 3) g(x) = 8x x 2 +12x + 6x 2 +18x + 9 g(x) = 8x x x + 9 Expand the equation, with each term separate. Expand. Gather like terms and simplify. Keep in mind, you may encounter hybrids of any or all of these formulas in a question, so try to remember as many of these as you can.

5 Example 6: Expand h(x) = (2x 4)4 (x 3)(x + 3) h(x) = (2x 4) 4 (x 3)(x + 3) Separate the function into two smaller equations, m(x) and n(x). m(x) = (2x 4) 4 n(x) = (x 3)(x + 3) Expand each mini equation. m(x) = (2x) C 1 (2x) 3 ( 4) + 4 C 2 (2x) 2 ( 4) C 3 (2x) 1 ( 4) 3 + ( 4) 4 m(x) = 16x 4 + (4)(8)( 4)x 3 + (6)(4)(16)x 2 + (4)(2)( 64)x m(x) = 16x 4 128x x 2 512x n(x) = (x + 3)(x 3) (a + b)(a b) = a 2 b 2 (x + 3)(x 3) = x 2 9 m(x)n(x) = h(x) Recombine the two expanded equations. Remember - m(x)n(x)=h(x) h(x) = (16x 4 128x x 2 512x + 256)(x 2 9) Expand the equation again, keeping each term seperate. h(x) = x 2 (16x 4 128x x 2 512x + 256) 9(16x 4 128x x 2 512x + 256) Expand and simplify. h(x) = 16x 6 128x x 4 512x x 2 144x x x x 2304 h(x) = 16x 6 128x x x x x 2304 Keep in mind, Example 6 probably goes a bit too far with the difficulty of the question. Don t expect something that crazy. What next? Now, you should go and attempt Self Help Tutorial - Expansion of Terms.