Topical Revision Worksheet 4 Polynomials Solution (SW)

Topical Revision Worksheet 4 Topic: Polynomials Question 1 [2008 EOY, Q11] marks (i) (a) Show that is a factor of . Solution: Sub : Hence is a factor of [1] (i) (b) Solve for x if . Solution: By long division, Hence or [4] (ii) The expression leaves a remainder of when divided by . Find the values of p and q. Solution: Sub , Sub , Hence and [5]

Question 2 [2010 EOY, Q12] marks (a) Solve for x when . Solution: By trial and error, when , hence is a factor. By long division, Hence, or [M1] [3] (b) It is given that where is a polynomial. (i) State the degree of . Solution: Degree of is 1 as it is linear [1] (ii) Find the values of a and b. Solution: Sub to get Sub to get Solve simultaneously to get and [3]

Question 3 [2011 EOY, Q11] marks (i) Given that has a factor and the other two roots of the equation are and . Find the value of and of . Solution: [4] (ii) Solve the equation . Solution: Found 1 linear factor Correct factorization into linear and quadratic e.g. Correct factorization into 3 linear factors [4]

Question 4 [2012 EOY, Q8] marks The cubic polynomial is such that the coefficient of is 2 and the roots of are −1, 2k and . It is given that has a remainder of 168 when divided by . i) Show that . Solution: Ability to express f as a product of three linear factors and a constant 2. [3] (ii) Hence find a value for k and show that there are no other real values of k which satisfy this equation. Solution: By trial and error, k = 3 Hence, there is only one real value of k. [4] (iii) Use your answer to part (ii) to solve the equation . Solution: [2]

Question 5 [2013 EOY, Q5] marks Given that is a factor of , find the value of a and of b. Solution: As Let x = 2 and subs into the eqn. Let and subs into the eqn. Solve the above two equations, a = 16 and b = 3 [4] Question 6 [2014 EOY, Q7] marks (i) Solve the equation . Solution: [4] (ii) Find the remainder when is divided by . Solution: [4]

Question 7 [2015 EOY, Q9] marks (i) Solve for x when . Solution: where is a root so Hence [5] (ii) Given that , where p and q are unknown constants, leaves a remainder of 2 when divided by and a remainder of 9 when divided by , find the values of p and q and hence find the remainder when is divided by . Solution: and Solving, and Hence By long division, remainder is [6]