This pdf includes the following topics:- Expanding brackets Simplifying expression Key points Examples Practice Problems
1. Expanding brackets and simplifying expressions A LEVEL LINKS Scheme of work: 1a. Algebraic expressions – basic algebraic manipulation, indices and surds Key points When you expand one set of brackets you must multiply everything inside the bracket by what is outside. When you expand two linear expressions, each with two terms of the form ax + b, where a ≠ 0 and b ≠ 0, you create four terms. Two of these can usually be simplified by collecting like terms. Example 1 Expand 4(3x − 2) 4(3x − 2) = 12x − 8 Multiply everything inside the bracket by the 4 outside the bracket Example 2 Expand and simplify 3(x + 5) − 4(2x + 3) 3(x + 5) − 4(2x + 3) 1 Expand each set of brackets = 3x + 15 − 8x – 12 separately by multiplying (x + 5) by 3 and (2x + 3) by −4 = 3 − 5x 2 Simplify by collecting like terms: 3x − 8x = −5x and 15 − 12 = 3 Example 3 Expand and simplify (x + 3)(x + 2) (x + 3)(x + 2) 1 Expand the brackets by multiplying = x(x + 2) + 3(x + 2) (x + 2) by x and (x + 2) by 3 = x2 + 2x + 3x + 6 = x2 + 5x + 6 2 Simplify by collecting like terms: 2x + 3x = 5x Example 4 Expand and simplify (x − 5)(2x + 3) (x − 5)(2x + 3) 1 Expand the brackets by multiplying = x(2x + 3) − 5(2x + 3) (2x + 3) by x and (2x + 3) by −5 = 2x2 + 3x − 10x − 15 = 2x2 − 7x − 15 2 Simplify by collecting like terms: 3x − 10x = −7x
2. 1 Expand. Watch out! a 3(2x − 1) b −2(5pq + 4q2) When multiplying (or c −(3xy − 2y2) dividing) positive and 2 Expand and simplify. negative numbers, if the signs are the same a 7(3x + 5) + 6(2x – 8) b 8(5p – 2) – 3(4p + 9) the answer is ‘+’; if the c 9(3s + 1) –5(6s – 10) d 2(4x – 3) – (3x + 5) signs are different the answer is ‘–’. 3 Expand. a 3x(4x + 8) b 4k(5k2 – 12) c –2h(6h2 + 11h – 5) d –3s(4s2 – 7s + 2) 4 Expand and simplify. a 3(y2 – 8) – 4(y2 – 5) b 2x(x + 5) + 3x(x – 7) c 4p(2p – 1) – 3p(5p – 2) d 3b(4b – 3) – b(6b – 9) 5 Expand 12 (2y – 8) 6 Expand and simplify. a 13 – 2(m + 7) b 5p(p2 + 6p) – 9p(2p – 3) 7 The diagram shows a rectangle. Write down an expression, in terms of x, for the area of the rectangle. Show that the area of the rectangle can be written as 21x2 – 35x 8 Expand and simplify. a (x + 4)(x + 5) b (x + 7)(x + 3) c (x + 7)(x – 2) d (x + 5)(x – 5) e (2x + 3)(x – 1) f (3x – 2)(2x + 1) g (5x – 3)(2x – 5) h (3x – 2)(7 + 4x) i (3x + 4y)(5y + 6x) j (x + 5)2 k (2x − 7)2 l (4x − 3y)2 9 Expand and simplify (x + 3)² + (x − 4)² 10 Expand and simplify. 2 1 2 1 a x x b x x x x
3. 1 a 6x – 3 b –10pq – 8q2 c –3xy + 2y2 2 a 21x + 35 + 12x – 48 = 33x – 13 b 40p – 16 – 12p – 27 = 28p – 43 c 27s + 9 – 30s + 50 = –3s + 59 = 59 – 3s d 8x – 6 – 3x – 5 = 5x – 11 3 a 12x2 + 24x b 20k3 – 48k c 10h – 12h3 – 22h2 d 21s2 – 21s3 – 6s 4 a –y2 – 4 b 5x2 – 11x c 2p – 7p2 d 6b2 5 y–4 6 a –1 – 2m b 5p3 + 12p2 + 27p 7 7x(3x – 5) = 21x2 – 35x 8 a x2 + 9x + 20 b x2 + 10x + 21 c x2 + 5x – 14 d x2 – 25 e 2x2 + x – 3 f 6x2 – x – 2 g 10x2 – 31x + 15 h 12x2 + 13x – 14 i 18x2 + 39xy + 20y2 j x2 + 10x + 25 k 4x2 − 28x + 49 l 16x2 − 24xy + 9y2 9 2x2 − 2x + 25 2 1 10 a x2 1 b x2 2 x2 x2
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