Test 8 1. Find the remainder if x3 + 4x2 - x + 3 is divided by 2x + 1. 2. Given: f(x) = 2x3 + x2 - 5x + 2. Factorise f(x) by using the factor theorem. 3. What must be added to the expression 2x - x + 1 so that 2x - 3 will be a factor? 4. Given: f (x) = 2x3 + ax + ax -2 If f(x) is divided by 2x + 1, the remainder is b. Determine a in terms of b. 5. Given: f(x) = x(x + p) - q and p # 0 and q # 0. and g(x) = 2x2 - q -rx. If (x - p) is a factor of f(x) and (x - q) is a factor of g(x), prove that r = 4p? - 1. 6. Given: g(x) = x + max + na x + 8a',a # 0. If (x - a)(x + 2a) is a factor of g(x), calculate the values of m and n. 7. Prove that x + 3y is a factor of a3 - 2x + 3yx2 - 6y. 8.1 For which value of m will x - 3 be a factor of 1+my -11x - 15m? 8.2 Hence deduce for which value(s) of m, (x - 3)(x + 2) will (x) be a factor of x+my - 11x - 15m. 9. If g(x) is divided by x + x - 6 the remainder is 7x + 13. Determine the remainder if g(x) is divided by x + 3. 10. Determine the smallest value of k for which the expression x3 + 5x - k(x -4) is divisible by X - k. 11. Solve for a in terms of y: x + 2xy- x12 - 23 =0 12. If p3 + p - 2 is a factor of p* - ap' - 5p + 8p - b, calculate a and b. 13. Solve for x if x = 12 - 3x + 2 2- x 14. Prove with the help of the remainder theorem that x+ 20 - 36 is a factor of x2 + 4ax + 4a- - 962. 15. If 2x3 - 1 1x - 5 = (2x - 1)(x+ 2) p(x) + ax+b. calculate 15.1 a and b 15.2 p(x), an expression in x. 16.1 Find an expression of the third degree divisible by (2x + 1) and (x - 2). The coefficient of is - I and the absolute term (constant term) is -2. 16.2 Write down the factors of the expression. 124