Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Algebra Worksheet 2 - Factorising 1. Factorise the following by taking out common factors: a) x 2 + 3x b) x 2 - x -3 c) 4a -1 + 6a 4 3x 4 x3 d) + 5 y 10 y 1 2 e) x - x f) a 1 2 3 2 +a -1 g) h) i) j) k) l) ab + ac + bd + cd 12bc + 35a 2 - 28ac - 15ab 3a 2 b(2s - t ) - 6ab(t - 2s) - 9ab 2 (2s - t ) 4 x 3 ( x - y) 2 - 6x 2 y ( x - y ) 3 (a + b) 2 - 3a - 3b 2 x( x - 2) + xy - 2 y(3y + 4) h) i) j) k) l) ( x + 5) 2 + 4( x + 5) + 3 (2a - 3) 2 - 8(2a - 3) + 16 2 g 2 h 2 - 8gh - 120 12 sin 2 x - 5 sin x cos x - 2 cos2 x 4 x - 23+x - 20 2 2. Factorise the following quadratic expressions: a) b) c) d) e) f) g) x 2 + 11x + 30 6x 2 - 84 x - 192 12 x 2 - x - 6 7 tan 2 x - 21 tan x - 196 cos3 x - 7 cos2 x + 12 cos x x 4 - 2 x 2 - 24 x 6 - 2 x 3 - 35 3. Factorise the following using common prime factor, where possible: a) b) c) d) e) f) 8x 2 + 14 x - 15 9 x 2 + 27 x + 20 50x 2 + 75x - 27 36x 2 + 105x + 49 200 x 2 + 30x - 27 121x 2 - 55x - 50 g) h) i) j) k) 27 x 2 + 66x - 80 600 x 2 - 90x - 27 28x 2 - 77 x - 72 250 x 2 + 35x - 98 216 x 2 + 30x + 5 (Sealy, Agnew; Senior Mathematics p9) – see rules at the end of this worksheet 4. Factorise: a) x 2 - y 2 b) 0.49 x 4 - 0.64 y 8 c) ( x + 3) 2 - ( x - 2) 2 d) (2 x + 3) 2 - ( x - 4) 2 e) 32 x - 4 x2 f) 1 25 3 2 g) 4 - 12 y x 1 h) x 2 - 2 + 2 x Delta: p 1-4, Ex 1.1-1.8 Mathematics with Calculus Page 1 JM Algebra Worksheet 2 - Factorising (Sidebotham; Mathematics Revision p5) Exercise 1.2 1. Factorise each of the following a. e. i. m. x 2 - x -12 x 2 - x - 20 6x 2 + 7x - 5 8z 2 + 2zy -15y 2 p. x 2 +11ax + 28a2 x 2 -16 3x 2 + 4x +1 25y 2 - 36 t4 - t2 16a 2 q. -1 b2 b. f. j. n. c. g. k. o. x 2 + 3x -18 4x 2 + 8x + 3 x 4 -16 t 4 -13t 2 + 36 d. a2bc - bc 2d h. 6x 2 +17x - 3 l. 4x 2 y - 2y 3 2. Factorise as far as possible a. ( x -1) - 2( x -1) +1 c. 2x 3 + x 2 - 8x - 4 e. 2a - 2b + a2 - b2 b. 1- (a + b)2 d. 3x - 3y + x 2 - xy f. 4x 2 (a - b) - (a - b) g. x 3 + 8 h. i. 64y 3 - 27 k. 125y 3 + 64 x 3 *m. 24 x -1 *j. ( x - y ) - 4(x - y)(x + y)2 l. sin4 q cos2 q + sinq cosq - 2 2 3 1 ( x +1) 2 - ( x +1) 2 3 3. Simplify each of the following by factorising a. c. x2 - 4 x 2 + 3x -10 x3 - 8 x2 - 4 1 a 2b + 2ab 2 ab 2 2x + 2 x +1 - 3 d. 2x + 3 b. 1 4(x - 3) 2 - (2x + 3)(x - 3) 2 e. x-3 Rules: Rule 1. (Most important) Always make sure that you have taken out all common factors first. 6 Rule 2. No horizontal row may contain a common factor. For example 32 -1 is not a possible arrangement, since the top row has a common factor of 3. Rule 3. If the middle term and one of the end terms have a common prime factor, then that prime factor must be a common factor of the two numbers in the corresponding vertical column. As 3 is a common factor of 18x 2 and - 3x in our example, then, 18 must be split into 63 with the common prime factor 3. Note: This rule applies to prime numbers. If 4 (a perfect square) is a common factor of the middle and one of the end terms, you can only be sure that the end term splits into two numbers with 2 as a common factor. However, the rule does work for a number like 6. Why? Rule 4. If a particular prime number p is a common factor of the middle term and an end term then p2 must also be a factor of that end term. Otherwise the trinomial has no real factors. 6x 2 + 3x - 28 has no real factors, because 3 is a common factor of the first two terms, but 9 is not a factor of 6. Mathematics with Calculus Page 2 JM