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9 Division and the remainder theorem A. Divide 2x 3 + x 2 − 6x + 9 by x + 2. B. Use the remainder theorem to find the remainder when 4x 3 − 2x 2 + 3x − 1 is divided by x − 2. 11 Properties of quadratic expressions and equations A. Solve the quadratic equations (i) x 2 − 3x = 0 (ii) x 2 − 5x + 6 = 0 (iii) 2x 2 + 3x − 2 = 0 B. Complete the square for the quadratic x 2 + x + 1 and hence determine its minimum value. C. What is the (i) sum (ii) product of the roots of the quadratic x 2 + 2x + 3? 13 The binomial theorem Expand (2 − x)5 by the binomial theorem.
A) If a : b = 7 : 3 determine a for the following values of b: (i) 4, (ii) 3, (iii) −7, (iv) 15. (b) Repeat for b with the same values for a. C. If a : b = 5 : 2 evaluate as fractions a b a b (v) + b a (i) a a+b a b (vi) − a+b a−b b a+b a+b a−b (vii) + a−b a+b (ii) (iii) a−b a+b (iv) D. 6 Factorial and combinatorial notation – permutations and combinations 4 16 18 ➤ (ii) ➤ (i) a when b = 3 ➤➤ A. Evaluate (i) 5! (iv) (ii) 10! 6! 16! (vii) 10! + 11! 14! )2 13 (v) 301! 300! 10! 6! (iii) 8! 9! 3! 4!
We use a to denote the positive value of the square root (although the notation has to be stretched when we get to complex numbers). For example 2= √ 4 since 22 = 4 √ √ Since −2 = −√ 4 also satisfies (−2)2 = 4, − 4 is also a square root of 4. So the square roots of 4 are ± 4 = ±2. We can similarly have √ cube roots of a number a, which yield a when they are cubed. If a is positive then 3 a denotes the positive value of the cube root. For example 2= √ 3 because 23 = 8 8 In the case of taking an odd root √ of a negative number the convention is to let denote the negative root value, as in 3 −8 = −2, for example.