1 x 0 y - 3 - 5 1 5 The diagram shows the graph of the cubic function ( ) y x f = . The intercepts of the curve with the axes are all integers. (a) Find the set of values of x for which ( )x 0 f 1 . [1] (b) Find an expression for ( )x f . [3]

1(a) 3 1 − x B1 1(b) ( )( )( ) 1 3 1 5 3 − + − − x x x 3 B1 for a negative cubic function B1 for a cubic function multiplied by 1 3 B1 for ( )( )( ) 3 1 5 + − − x x x (a) To earn this mark, you must correctly identify the range of x values where the function is less than or equal to zero, including the endpoints. (b) To earn these marks, you need to show that the function is a negative cubic, that it has a factor of 1/3, and that the factors (x+3), (x-1), and (x-5) are present.

3 The polynomial ( ) p x ax x bx 9 6 3 2 = - + - , where a and b are constants, has a factor of x 2 - . The polynomial has a remainder of 66 when divided by x 3 - . (a) Find the value of a and of b. [4] (b) Using your values of a and b, show that ( ) ( ) ( ) p q x x x 2 = - , where ( ) q x is a quadratic factor to be found. [2] (c) Hence show that the equation ( ) p x 0 = has only one real solution. [2]

3(a) ( ) p 2 : 8 36 2 6 0 − + − = a b B1 ( ) p 3 : 27 81 3 6 66 − + − = a b B1 M1 Dep on at least one of the previous B marks, for attempt to solve their equations and obtain a solution for both a and b 6, 3 = = − a b A1 For both 3(b) ( )( ) 2 2 6 3 3 − + + x x x 2 M1 for attempt at quadratic factor either by observation to obtain 2 6 3 + + x px or by algebraic long division to obtain at least 2 6 3 ... + x x A1 all correct 3(c) Discriminant of ( ) q x = 2 3 4 6 3 − × × 63 = − which is < 0 M1 For calculation of discriminant and confirmation that it is < 0 ( ) q 0 = x has no real solutions hence ( ) p 0 = x has only one real solution A1 For a correct conclusion from correct work. (a) To earn the first two marks, you must correctly set up the equations for p(-2) and p(3) based on the given remainders. The third mark requires you to attempt to solve these two equations to find values for 'a' and 'b', and the final mark is for correctly stating both 'a' and 'b'. (b) To earn the first mark, you need to show an attempt to find the quadratic factor, either by observation or algebraic long division, resulting in a quadratic expression with the correct x^2 and x terms. The second mark is for correctly stating the complete quadratic factor. (c) To earn the first mark, you must calculate the discriminant of the quadratic factor q(x) and correctly conclude that it is less than zero. The second mark is for using this information to correctly conclude that p(x)=0 has only one real solution.

6 DO NOT USE A CALCULATOR IN THIS QUESTION. The polynomial ( )x x ax x b 10 10 p 3 2 = + - + , where a and b are integers, is divisible by x 2 1 + . When ( )x p is divided by x 1 + , the remainder is 24 - . (a) Find the value of a and of b. [4] (b) Find an expression for ( )x p as the product of three linear factors. [4] (c) Write down the remainder when ( )x p is divided by x. [1]

6(a) 5 0 5 4 4 = − + + + a b M1 For use of the factor theorem 24 10 10 − = − + + + a b M1 For use of the remainder theorem 4 15 + = − a b 24 + = − a b leading to M1 Dep on both previous M marks for solution of their equations without using a calculator 27 = − a , 3 = b A1 6(b) ( )( ) 2 2 1 5 ........... + + x x their b M1 Allow for observation or algebraic long division. Their a and b must be integers. ( )( ) 2 2 1 5 16 3 + − + x x x A1 ( )( )( ) 2 1 5 1 3 + − − x x x 2 M1 for attempt to factorise their 3-term quadratic A1 all correct from fully correct working 6(c) 3 B1 FT on their (integer) b (a) To earn these marks, you must correctly apply the factor theorem for (x+1/2) and the remainder theorem for (x+1) to set up two simultaneous equations. Then, you must correctly solve these equations without a calculator to find the values of a and b. (b) To earn these marks, you must first use the factor (x+1/2) to find the quadratic factor, either by observation or algebraic long division. Then, you need to correctly factorize this quadratic into two linear factors to express the polynomial as a product of three linear factors. (c) To earn this mark, you simply need to state the value of 'b' as the remainder when p(x) is divided by x.

4 The polynomial ( ) p x mx x x n 29 39 3 2 = - + + , where m and n are constants, has a factor x 3 1 - , and remainder 6 when divided by x 1 - . Show that x 2 - is a factor of ( ) p x . [6]

4 29 39 0 27 9 3 − + + = m n oe B1 m – 29 + 39 + n = 6 oe B1 Eliminates one unknown correctly for a pair of linear equations in m and n and solves for one unknown M1 m = 6, n = −10 A2 A1 for either [p(2) =] 48 – 116 + 78 – 10 = 0 oe, nfww A1 To earn marks, you must correctly substitute the factor and remainder information into the polynomial to form two equations, then solve these equations for m and n. Finally, substitute x=2 into the polynomial and show that the result is 0.

9 (a) Find the equation of the normal to the curve y x x x 4 6 3 2 = + - + at the point ( , ) 1 4 . [5] (b) DO NOT USE A CALCULATOR IN THIS PART OF THE QUESTION. Find the exact x-coordinate of each of the two points where the normal cuts the curve again. [5]

9(a) 2 d 3 2 4 d y x x x = + − M2 M1 for any two terms correct d 1 1 d y x x = → = A1 [ ] m⊥= −1 M1 FT 1 1 their − y – 4 = −1(x – 1) oe isw A1 FT their m⊥ 9(b) ( ) 3 2 4 6 5 x x x their x + − + = −+ [ ] 3 2 3 1 0 x x x → + − + = M1 FT their linear equation of the form y = mx + c where m ≠ 0 and c ≠ 0 from (a) Correct quadratic factor: x2 + 2x – 1 B2 B1 for any two out of three terms correct Must be from the correct cubic Solves their (x2 + 2x – 1) = 0 using the formula or by completing the square M1 dep on M1 and valid attempt at finding quadratic factor M0 if their quadratic factor does not have real roots 2 8 2 −± isw or 2 2 2 2 −± isw A1 (a) To earn marks, you must correctly differentiate the curve equation to find the gradient at the given point, then find the negative reciprocal of this gradient to get the gradient of the normal, and finally use the point-gradient form to write the equation of the normal. (b) You earn marks by setting the equation of the normal equal to the equation of the curve, rearranging to form a cubic equation, identifying (x-1) as a factor, and then solving the resulting quadratic factor using the quadratic formula to find the exact x-coordinates.

Cambridge O Level Additional Mathematics · 4037

O-Level Additional Mathematics (4037) Factors of polynomials past paper questions

Revise O-Level Additional Mathematics (4037) — Factors of polynomials with real Cambridge past-paper questions grouped by topic, each shown with its marking scheme. Working one topic at a time targets exactly the skill each question tests.

34 free questions from 2016–2021, each with its marking scheme. Open the full topical bank to filter by subtopic, paper, and year.

Factors of polynomials in O-Level Additional Mathematics 4037 exams, 2020–2025

Across 2020–2025, Factors of polynomials accounted for 39 questions worth 267 marks in Cambridge O Level Additional Mathematics (4037) papers.