Core Mathematics 2 Algebra Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Algebra 1
Algebra and functions Simple algebraic division; use of the Factor Theorem and the Remainder Theorem. Only division by (x + a) or (x a) will be required. Students should know that if f(x) = 0 when x = a, then (x a) is a factor of f(x). Students may be required to factorise cubic expressions such as x 3 + 3x 2 4 and 6x 3 + 11x 2 x 6. Students should be familiar with the terms quotient and remainder and be able to determine the remainder when the polynomial f(x) is divided by (ax + b). Core Mathematics 2 Algebra 2
Simplifying Algebraic Fractions Algebraic fractions can be simplified by division. Example 1: Simplify 3x3 + 5x 2 2x x = 3x3 x + 5x 2 x 2x x = 3x 2 + 5x 2 Example 2: Simplify 6x4 5x 3 + 2x 2 2x = 6x4 2x 5x 3 2x + 2x 2 2x = 3x 3 + 5x2 2 x Factorising to Cancel Sometimes you need to factorise before you can simplify Example 1: Simplify x2 + 7x + 12 x + 3 = (x + 3)(x + 4) = x + 4 x + 3 Example 2: Simplify 2x2 + 5x 12 2x 2 7x + 6 = (2x 3)(x + 4) (x 2)(2x 3) = x + 4 x 2 Core Mathematics 2 Algebra 3
Dividing a Polynomial by (x±p) You divide polynomials in the same way as you perform long division. Example 1: Divide 6x 3 + 28x 2 7x + 15 by (x + 5) 6x 2 2x + 3 = (x + 5) 6x 3 + 28x 2 7x + 15 6x 3 + 30x 2 2x 2 7x 2x 2 10x 3x + 15 3x + 15 0 6x3 + 28x 2 7x + 15 x + 5 = 6x 2 2x + 3 Example 2: Divide 5x 3 27x 2 + 23x + 30 by (x + 6) 5x 2 + 3x + 5 = (x + 6) 5x 3 27x 2 + 23x + 30 5x 3 30x 2 3x 2 + 23x 3x 2 + 18x 5x + 30 5x + 30 0 5x3 27x 2 + 23x + 30 x + 6 = 5x 2 + 3x + 5 Core Mathematics 2 Algebra 4
Finding the Remainder After a division of a polynomial, if there is anything left then it is called the remainder. The answer is called the quotient. If there is no remainder then what you are dividing by is said to be a factor. Example 1: Divide 2x 3 5x 2 16x + 10 by (x 4) and state the remainder and quotient 2x 2 + 3x 4 = (x 4) 2x 3 5x 2 16x + 10 2x 3 8x 2 3x 2 16x 3x 2 12x 4x + 10 4x + 16 6 The remainder is 6 and the quotient is 2x 2 + 3x 4 Example 2: Divide 2x 3 + 9x 2 + 25 by (x + 5) and state the remainder and quotient 2x 2 x + 5 = (x + 5) 2x 3 + 9x 2 + 0x + 25 2x 3 + 10x 2 x 2 + 0x x 2 5x 5x + 25 5x + 25 0 Remember to use 0x as there is no x term. The remainder is 0 and the quotient is 2x 2 x + 5 Core Mathematics 2 Algebra 5
Factor Theorem Notes: Factor Theorem: (x a) is a factor of a polynomial f(x) if f(a) = 0. Remainder Theorem: The remainder when a polynomial f(x) is divided by (x a) is f(a). Extended version of the factor theorem: b (ax + b) is a factor of a polynomial f(x) if f 0. a To find out the factors of a polynomial you can quickly substitute in values of x to see which give you a value of zero. Example 1: Given f(x) = x 3 + x 2 4x 4. Use the factor theorem to find a factor f(x) = x 3 + x 2 4x 4 f(1) = (1) 3 + (1) 2 (4 1) 4 = 6 (x 1) is not a factor f = 3 + 2 (4 2) 4 = 0 (x 2) is a factor To go from the x value to the factor simply put it into a bracket and change the sign. Remember this:- (x + 3)(x 2) = 0 x = 3 and x = 2 We are now going in reverse by putting the x value back into the brackets Core Mathematics 2 Algebra 6
Example 2: Given f(x) = 3x 2 + 8x 2 + 3x 2. Factorise fully the given function f(x) = 3x 3 + 8x 2 + 3x 2 f(1) = 3 (1) 3 + 8 (1) 2 + (3 1) 2 = 12 (x 1) is not a factor f = 3 3 + 8 2 + (3 2) 2 = 60 (x 2) is not a factor f( 1) = 3 ( 1) 3 + 8 ( 1) 2 + (3 1) 2 = 0 (x + 1) is a factor To fully factorise it we now need to find the quotient. We do this by dividing by the factor we have just found. 3x 2 + 5x 2 = (x + 1) 3x 3 + 8x 2 + 3x 2 3x 3 + 3x 2 5x 2 + 3x 5x 2 + 5x 2x 2 2x 2 0 (x + 1)(3x 2 + 5x 2) = (x + 1)(3x 1)(x + 2) We have factorised the quotient further to get the fully factorised answer Core Mathematics 2 Algebra 7
Using the Factor Theorem to find the Remainder If the polynomial f(x) is divided by (ax b) then the remainder is f b a Example 1: Find the remainder when f(x) = 4x 4 4x 2 + 8x 1 is divided by (2x 1) Using (ax b) then a = 2, b = 1 so f 1 2 f(x) = 4x 4 4x 2 + 8x 1 f 1 2 = 4 1 4 4 2 1 2 + 2 8 1 2 1 = 4 16 4 4 + 8 2 1 = 1 4 1 + 4 1 = 2 1 4 Remainder is 2 1 4 Example 2: Find the remainder if f(x) = x 3 20x + 3 is divided by (x 4) Using (ax b) then a = 1, b = 4 so f f(x) = x 3 20x + 3 f = 3 (20 4) + 3 = 64 80 + 3 = 13 Remainder is 13 Core Mathematics 2 Algebra 8
(C2-1.1) Name: Homework Questions 1 Simplifying Algebraic Fractions Simplify the following by division a) 3x 3 + 2x 2 + x x b) 5x 3 + 4x 2 2x 2x c) 4x 3 + 3x 2 + 7x x d) 8x 4 + 6x 3 6x 2 2x 2 e) 5x 2 + 6x 3 x 2 f) 7x 3 3x 2 + 5x x 2 g) 3x 5 6x 3 2x 2 h) 7x 5 + 3x 2 + 6 3x 2 i) 4x 2 + 6x 5 2x j) 12x 3 9x 2 + 3x 3x 2 Core Mathematics 2 Algebra 9
(C2-1.2) Name: Homework Questions 2 Factorising to Cancel Simplify the following by factorizing first a) x 2 + 5x + 6 x + 2 b) x 2 + 6x 7 x 1 c) x 2 + 2x 15 x 3 d) x 2 + 10x + 24 x + 6 e) 2x 2 10x + 12 x 3 f) x 2 9x + 20 x 4 g) x 2 49 x 7 h) x 2 + 10x + 9 x + 1 i) 4x 2 + 12x + 9 2x + 3 j) 7x 3 y 28xy 3 x 2y Core Mathematics 2 Algebra 10
(C2-1.3) Name: Homework Questions 3 Dividing a Polynomial Divide the following to find the quotient a) 2x 3 + 3x 2 3x 2 by (x + 2) b) 3x 3 2x 2 19x 6 by (x 3) c) x 3 + 7x 2 6x 72 by (x 3) e) 4x 3 + 3x 2 25x 6 by (x 2) Core Mathematics 2 Algebra 11
(C2-1.4) Name: Homework Questions 4 Finding the Remainder Find the remainder by division when the following polynomials are divided by:- a) x 2 4x + 5 by (x 1) b) 2x 3 + 3x 2 3x + 2 by (x + 2) c) 2x 2 + 7x 3 by (2x 1) d) x 3 2x 2 7x 1 by (x + 1) Core Mathematics 2 Algebra 12
(C2-1.5) Name: Homework Questions 5 Factor Theorem Use the factor theorem to show that the following are factors, and hence fully factorise. a) (x 3) is a factor of x 3 3x 2 4x + 12 b) (x + 2) is a factor of x 3 3x 2 6x + 8 c) (x + 1) is a factor of x 3 + 8x 2 + 19x + 12 d) (x 2) is a factor of x 3 4x 2 11x + 30 e) (x + 3) is a factor of x 3 + 2x 2 5x 6 Core Mathematics 2 Algebra 13
(C2-1.6) Name: Homework Questions 6 Finding Remainder Using Factor Theorem Find the remainder when the following polynomial is divided by:- a) 3x 3 2x 2 + 5x 12 by (x 1) b) 7x 3 5x 2 + 12x + 16 by (x + 2) c) 6x 3 + 4x 2 9x + 10 by (x + 1) d) 5x 5 4x 3 + 2x 2 8x + 11 by (x + 3) e) 7x 4 + 7x 3 6x 2 + 9x + 2 by (x 2) f) 5x 3 5x 2 + 9x + 12 by (x + 1) g) 7x 4 4x 3 + 5x 2 6x + 15 by (x + 2) h) 4x 3 + 6x 2 + 3x + 4 by (x 3) i) 7x 4 9x 3 8x 2 7x + 5 by (x 1) j) 6x 5 + 4x 3 + 2x 2 x 4 by (x 3) Core Mathematics 2 Algebra 14
Past examination questions. 1. f(x) = x 3 2x 2 + ax + b, where a and b are constants. When f(x) is divided by (x 2), the remainder is 1. When f(x) is divided by (x + 1), the remainder is 28. (a) Find the value of a and the value of b. (6) (b) Show that (x 3) is a factor of f(x). (C2, Jan 2005 Q5) 2. (a) Use the factor theorem to show that (x + 4) is a factor of 2x 3 + x 2 25x + 12. (b) Factorise 2x 3 + x 2 25x + 12 completely. 3. f(x) = 2x 3 + x 2 5x + c, where c is a constant. Given that f(1) = 0, (C2, June 2005 Q3) (a) find the value of c, (b) factorise f(x) completely, (c) find the remainder when f(x) is divided by (2x 3). 4. f(x) = 2x 3 + 3x 2 29x 60. (a) Find the remainder when f(x) is divided by (x + 2). (b) Use the factor theorem to show that (x + 3) is a factor of f(x). (c) Factorise f(x) completely. 5. f(x) = x 3 + 4x 2 + x 6. (C2, Jan 2006 Q1) (C2, May 2006 Q4) (a) Use the factor theorem to show that (x + 2) is a factor of f(x). (b) Factorise f(x) completely. Core Mathematics 2 Algebra 15
(c) Write down all the solutions to the equation x 3 + 4x 2 + x 6 = 0. (1) (C2, Jan 2007 Q5) 6. f(x) = 3x 3 5x 2 16x + 12. (a) Find the remainder when f(x) is divided by (x 2). Given that (x + 2) is a factor of f(x), (b) factorise f(x) completely. (C2, May 2007 Q2) 7. (a) Find the remainder when x 3 2x 2 4x + 8 is divided by (i) x 3, (ii) x + 2. (b) Hence, or otherwise, find all the solutions to the equation x 3 2x 2 4x + 8 = 0. 8. f(x) = 2x 3 3x 2 39x + 20 (a) Use the factor theorem to show that (x + 4) is a factor of f (x). (b) Factorise f (x) completely. (3) (C2, Jan 2008 Q1) (C2, June 2008 Q1) 9. f (x) = x 4 + 5x 3 + ax + b, where a and b are constants. The remainder when f(x) is divided by (x 2) is equal to the remainder when f(x) is divided by (x + 1). (a) Find the value of a. Core Mathematics 2 Algebra 16 (5)
Given that (x + 3) is a factor of f(x), (b) find the value of b. (3) (C2, Jan 2009 Q6) 10. f(x) = (3x 2)(x k) 8 where k is a constant. (a) Write down the value of f (k). (1) When f(x) is divided by (x 2) the remainder is 4. (b) Find the value of k. (c) Factorise f (x) completely. (3) (C2, June 2009 Q3) 11. f(x) = 2x 3 + ax 2 + bx 6, where a and b are constants. When f(x) is divided by (2x 1) the remainder is 5. When f(x) is divided by (x + 2) there is no remainder. (a) Find the value of a and the value of b. (b) Factorise f(x) completely. (6) (3) (C2, Jan 2010 Q3) 12. f(x) = 3x 3 5x 2 58x + 40. (a) Find the remainder when f (x) is divided by (x 3). Given that (x 5) is a factor of f(x), (b) find all the solutions of f(x) = 0. (5) (C2, June 2010 Q2) Core Mathematics 2 Algebra 17
13. f(x) = x 4 + x 3 + 2x 2 + ax + b, where a and b are constants. When f(x) is divided by (x 1), the remainder is 7. (a) Show that a + b = 3. When f(x) is divided by (x + 2), the remainder is 8. (b) Find the value of a and the value of b. (5) (C2, Jan 2011 Q1) 14. f(x) = 2x 3 7x 2 5x + 4 (a) Find the remainder when f(x) is divided by (x 1). (b) Use the factor theorem to show that (x + 1) is a factor of f(x). (c) Factorise f(x) completely. (C2, May 2011 Q1) 15. f(x) = x 3 + ax 2 + bx + 3, where a and b are constants. Given that when f (x) is divided by (x + 2) the remainder is 7, (a) show that 2a b = 6. Given also that when f(x) is divided by (x 1) the remainder is 4, (b) find the value of a and the value of b. (C2, Jan 2012 Q5) 16. f(x) = 2x 3 7x 2 10x + 24. (a) Use the factor theorem to show that (x + 2) is a factor of f(x). (b) Factorise f(x) completely. (C2, May 2012 Q4) Core Mathematics 2 Algebra 18
17. f(x) = ax 3 + bx 2 4x 3, where a and b are constants. Given that (x 1) is a factor of f(x), (a) show that a + b = 7. Given also that, when f(x) is divided by (x + 2), the remainder is 9, (b) find the value of a and the value of b, showing each step in your working. 18. f(x) = 2x 3 5x 2 + ax + 18 (C2, Jan 2013 Q2) where a is a constant. Given that (x 3) is a factor of f(x), (a) show that a = 9, (b) factorise f(x) completely. Given that g(y) = 2(3 3y ) 5(3 2y ) 9(3 y ) + 18, (c) find the values of y that satisfy g(y) = 0, giving your answers to 2 decimal places where appropriate. (3) (C2, May 2013 Q3) 19. f(x) = 4x 3 + ax 2 + 9x 18, where a is a constant. Given that (x 2) is a factor of f(x), (a) find the value of a, (b) factorise f(x) completely, (c) find the remainder when f(x) is divided by (2x 1). (3) (C2, May 2014 Q4) Core Mathematics 2 Algebra 19
20. f(x) = 2x 3 7x 2 + 4x + 4. (a) Use the factor theorem to show that (x 2) is a factor of f(x). (b) Factorise f(x) completely. (C2, May 2014 Q2) 21. f(x) = 6x 3 + 3x 2 + Ax + B, where A and B are constants. Given that when f(x) is divided by (x + 1) the remainder is 45, (a) show that B A = 48. Given also that (2x + 1) is a factor of f(x), (b) find the value of A and the value of B. (c) Factorise f(x) fully. (3) (C2, May 2015 Q3) Core Mathematics 2 Algebra 20