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Bracket Expansion and Factorisation

Bracket Expansion and Factorisation. Slideshow 15 Mathematics Mr Sasaki Room 307. Objectives. To recall how to expand pairs of brackets for a quadratic To be able to factorise quadratics in the form 2 + b + c To be able to solve quadratics in the form 2 + b + c = 0. Definitions.

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Bracket Expansion and Factorisation

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  1. Bracket Expansion and Factorisation Slideshow 15 Mathematics Mr Sasaki Room 307

  2. Objectives • To recall how to expand pairs of brackets for a quadratic • To be able to factorise quadratics in the form 2 + b + c • To be able to solve quadratics in the form 2 + b + c = 0.

  3. Definitions Today, we are dealing with a certain form of polynomial. Each has a special name. This is a “constant”. It doesn’t change. It’s also a monomial (one term). 4 4 + 3 This is “linear”. 42 + 3 - 2 This is a “quadratic”. 23 - 42 + 3 - 2 This is a “cubic”. 4 + 23 - 42 + 3 - 2 This is a “quartic”. 75 + 4 +23 - 42 + 3 - 2 This is a “quintic”.

  4. Expanding Brackets To expand a pair of brackets representing a quadratic, we multiply each term inside each bracket by each term in the other bracket. Here are the combinations. Notice that ab and cd are not combinations.

  5. Expanding brackets Try the example below. Example Try the worksheet! Expand (2 – 1)(4 + 6). (2-1)(4+6) 4 12 82 - - 6 = + = 82+ 8- 6

  6. Answers

  7. Factorisation Placing a quadratic into a pair of brackets is called “factorisation”. This is the opposite of expanding brackets and more difficult to do. Let’s try a linear expression. Example 3 What is the largest factor that divides into 9 and 6? Factorise 9 – 6. 3() 9- 6 3- 2 = The contents of the bracket is divided by the coefficient outside.

  8. Factorisation A quadratic is more difficult. We need to think of two numbers which add together to make 5 and multiply to make 6. Example Factorise + 5x + 6. 2 and 3 +5+ 6 Each bracket contains . 2 and 3 are positive so we get + 2 and + 3. The term has a coefficient of 1 because has a coefficient of 1. () () + 2 + 3 = If you are unsure it’s right, expand it out to check!

  9. Factorisation Let’s try another example. We need to think of two numbers which add together to make -5 and multiply to make -36. Example Factorise - 5x - 36. ()() + 4 - 9 = Hint: 9 – 4 is 5 and 9 x 4 is 36. -9 and 4 We will only look at quadratic expressions where the coefficient of is 1. -9 + 4 = -5-9 x 4 = -36 Try the worksheet!

  10. Answers

  11. Solving quadratic equations through factorisation We now know how to factorise quadratics. But how do we solve them for f() = 0? [f() means a function of .] Example Solve - 6x + 5 = 0. ()()= 0 - 5 - 1 This means that – 5 = 0 and – 1 = 0. So = 5 or = 1.

  12. Solving quadratic equations through factorisation Example Solve + 18x + 72 = 0. ()()= 0 + 6 + 12 So . Try the last worksheet!

  13. Answers or or or 7 or

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