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C3 Chapter 1 Algebraic Fractions

C3 Chapter 1 Algebraic Fractions. Dr J Frost (jfrost@tiffin.kingston.sch.uk). Last modified: 13 th May 2014. RECAP : Terminology. ?. dividend. dividend. quotient. ?. quotient. ?. remainder. remainder. 7. 1. =. 2 +. 3. 3. divisor. ?. divisor. RECAP : Algebraic Fractions. ?.

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C3 Chapter 1 Algebraic Fractions

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  1. C3 Chapter 1 Algebraic Fractions Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 13th May 2014

  2. RECAP: Terminology ? dividend dividend quotient ? quotient ? remainder remainder 7 1 = 2 + 3 3 divisor ? divisor

  3. RECAP: Algebraic Fractions ? Simplify the following fractions. 1 ? 6 ? 2 ? 7 ? 3 ? 8 ? 4 Bro Tip: When you have a fraction within a fraction, multiply the top and bottom of the outer fraction by the denominator of the inner one. ? 5

  4. Test Your Understanding Simplify: Express as a single fraction: ? ? Bro Tip: We can’t simplify this because the numerator and denominator don’t have a common factor we can divide by. is not in general divisible by !! ? ?

  5. Exercises Exercise 1A Exercise 1C 1a ? 1j ? ? 1b ? 1k ? 1g ? 1c ? ? 1o 1i ? 1g Exercise 1B ? ? 1k 1g ? ? 1l 1i ? 1n ? 1j ? 1o ? 1l ? 1q ? 1n

  6. RECAP: Algebraic Division 6x2 + 3 - 2x 6x3 + 28x2 – 7x + 15 x + 5 6x3+ 30x2 – 2x2 – 7x The Anti-Idiot Test: You can check your solution by expanding (x+5)(6x2 – 2x + 3) – 2x2 – 10x 3x + 15 3x + 15 0

  7. Test Your Understanding 2x2 + 3x – 4 2x3 – 5x2 – 16x + 10 x - 4 2x3– 8x2 Find the remainder. 3x2 – 16x 3x2 – 12x -4x + 10 Q: Is (x-4) a factor of 2x3 – 5x2 – 16x + 10? -4x + 16 -6

  8. Alternative Method: Remainder Theorem RECAP When a polynomial is divided by the remainder is . When a polynomial is divided by the remainder is . When a polynomial is divided by the remainder is When a polynomial is divided by the remainder is ? ? ? ? Given that , we could say that: ! Similarly, for a polynomial : ? It’s an identity (using ) because the equality is true for any value of .

  9. Alternative Method: Remainder Theorem Q As before, divide by but now using the remainder theorem. So We can compare coefficients of , and constant terms each side to find : We didn’t really need the last equation. But we get . Therefore: ? Comparing terms… Comparing terms… Comparing terms… Comparing constant terms… (Recall that )

  10. Test Your Understanding Q Divide by by using the Remainder Theorem. ? So Comparing coefficients: Thus

  11. Exercise 1D Express the following improper fractions in ‘mixed’ number form by: i) Using long division ii) Using the remainder theorem. ? a A cleverer way to divide is to note that is the difference of two cubes. thus . You do not need to know this factorisation at A Level. ? c ? e ? g Again, a simpler method using difference of two squares: ? h

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