Worksheets for GCSE Mathematics. Algebraic Expressions. Mr Black 's Maths Resources for Teachers GCSE 1-9. Algebra

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1 Worksheets for GCSE Mathematics Algebraic Expressions Mr Black 's Maths Resources for Teachers GCSE 1-9 Algebra

2 Algebraic Expressions Worksheets Contents Differentiated Independent Learning Worksheets Collecting Like Terms Algebraic Products Substituting into Expressions Expanding Algebraic Expressions Factorising Expressions Product of Two Brackets Factorising Quadratics when a = 1 Factorising Quadratics when a 1 Page 30 Page 40 Page 5 Page 60 Page 70 Page 80 Page 90 Page 100 Solutions Collecting Like Terms Algebraic Products Substituting into Expressions Expanding Algebraic Expressions Factorising Expressions Product of Two Brackets Factorising Quadratics when a = 1 Factorising Quadratics when a 1 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18

3 Collecting Like Terms Q1. Simplify these expressions by collecting like terms. a) aa + aa + aa + aa b) rr + rr + rr + gg + gg c) 4ee + ee + 2aa aa d) 4ff + 5ww + 3ff ww e) 5h + 8uu + 5 uu f) 6xx xx + 8 g) 2ww + 6bb + 3ww 4bb + 2 h) 4dd + 3ee 3dd 3ee i) 4 + 3vv 2 + 5vv + aa j) 3rr rr rr 4 k) 3dd + 3ff ff 3dd 2ff l) 12 + tt + 6gg gg Q2. Expand these pairs of brackets: a) aaaa + 5aaaa b) 4ffff + ffff + ffff c) aaaaaa + 5aaaaaa 2bbbbbb d) 2xx 2 + 4xx 2 e) 5xx 2 + 4rr 2 xx 2 f) 6dd 2 + dd dd 2 + 4dd g) 6yy gg 2 + 9gg h) 2mm 2 + xx 2 2mm 2 + 4xx 2 i) yy 2 + 3yy 2 + yy + yy 2 + 4yy j) aaaa 2 + aaaa 2 + 2bb + 2aa k) 4rr + rr rr + 2rr 2 l) 4tt + 3tt tt + 2jj 2 Q3. In each of these pyramids the cell above is the cell of the two directly beneath it. Complete the pyramids by collecting the like terms. Q4. Write the perimeter of each shape using a simplifying algebraic expression. a) b) 2

4 Algebraic Products Q1. Simplify these expressions. a) 3 aa b) 5 rr c) 7 2 rr d) rr ww e) h ee 3 f) 8 uu ii 2 g) 2dd 3 h) 4dd 6 i) 2 3gg 6 j) 2 rr ee k) 4 qq xx 2dd l) 9 ww 3ee 2ss Q2. Simplify these expressions. a) yy yy b) 2xx xx c) 4 rr 2rr d) h ff 2h ff e) 2pp 3qq f) 2xxxx 3xxxx g) ii 2ii 3rr h) 8mm mm 2nn i) 6gg 2ff gg aa j) 8yy xx 2ee xxxx k) 4gg ee 2gg 2eeee l) 4rrrr 6ffff ffff Q3. The top of each shape is the sum of the two expressions and the bottom is the product. Complete the missing terms in each cell. The first one has been done for you. Q4. Investigate possible perimeters for these rectangles given their fixed area: a) b) 3

5 Substituting into Expressions Q1. Substitute these values into the following expressions: a = 4 b = 2 c = 5 d = 8 a) aa + 12 b) bb + 5 c) cc 4 d) 5dd e) 9aa f) 2bb + 6 g) cc 5 h) dd 2 i) 6aa 4 j) dd k) 2cc + aa l) aaaa 2 Q2. Substitute these values into the following expressions: gg = 1 h = 7 xx = 9 ww = 10 a) 2gg + ww b) 5xx h c) gggg + h d) 6hww e) 3ww 8gg f) 2gg + hww g) ww 5gg h) h+xx 4 i) xx h 2 j) h+3 ww + 4 k) hxx gg l) hxxxx + 4gg Q3. Use these values to match the following expressions: uu = 3 tt = 6 mm = 12 dd = 20 2dd + 6 4tt + uu 5uu dd + mm tt dd + tt + 1 uuuu tt 3mm tt 4 uuuu 11 Q4. Use these values to place the following expressions in ascending order. rr = 1 tt = 10 mm = 4 dd = 0.5 2tt mmmm 4mm + rr 3tt + rrrr dddd + 5mm 4

6 Expanding Algebraic Expressions Q1. Expand these pairs of brackets: a) 3(xx + 2) b) 4(zz + 6) c) 6(4 + uu) d) 7(xx 4) e) 4(aa + 5) f) 10(cc 1) g) 2(5 yy) h) 8(1 + gg + h) i) 3(ff rr) j) 4(2ee + 5) k) 9(4 tt + 2ee) l) 7(6yy + 2 3tt ) Q2. Expand these pairs of brackets: a) xx(xx + 1) b) ww(ww 1) c) tt(3 + tt) d) gg(3gg + 2) e) h(2xx h) f) 2dd(5 6rr) g) 5tt(2tt 5rr) h) 4jj(xx + 2) i) 8uu(xx 5uu) j) 2ss(4 ss + 2uu) k) 5bb(bb 6yy + 5ee) l) 3mm(h + 4gg + 3mm) Q3. Expand and simplify each expression. a) 3(xx + 1) + 2(xx + 4) b) 4(gg + 3) + 3(gg + 5) c) 5(aa + 6) + 4(aa 2) d) 6(cc + 10) + 2(5 + cc) e) 3(3 ff) 2 (ff + 4) f) 6(3xx + 1) + 2(3 xx) g) 3(4yy 8) 2 (1 2yy) h) 4(6 5tt) (1 + tt) i) 3(rr + 1) 3(5 6rr) Q4. Write an expression, in terms of xx for the shaded area of each shape. 5

7 Factorising Expressions Q1. Fully factorise each of these expressions: a) 4h + 12 b) 2aa + 8 c) 7tt 14 d) 8yy 24 e) 12gg + 6 f) 8yy 4 g) 5uu + 10tt + 15 h) 4ww 8uu + 12tt i) 6tt + 3bb 9ii j) 3rr kk k) 32ee 4 l) 42jj + 7rr Q2. Expand these pairs of brackets: a) aaaa + aaaa b) rrrr 3rr c) yyyy 3gg d) 6tttt + 5ffh e) 12gggg + 18gggg f) 44uuh + 11tttt + 33uu g) aaaa + bb 2 h) 4cccc + 8ff 2 i) ww + ww 2 j) xx xx 2 k) 3cc 2 + 6cc l) 5aaaa aaaa Q3. Match together the equivalent expressions: 4(5 2xx) 4xx 2 4xx 4(5 + 6xx) 4xx 2 12rrrr 16xx xx xx 4xx(4xx + 5) 4xx(xx 3rr) 20 8xx 4xx(xx 1) 5(2tt 2 5) 5tt (3 + 5tt) 10tt 2 25 tt 2 + 5tt 5(tt 2 5) tt(tt + 5) 2tt 2 2tt tt 2tt(tt 1) Q4. Which of the following is the correct full factorisation of 48xxyy 2 12xx 2 yy? a) 12xx(4yy 2 3xxxx) b) 12yy(4xxxx xx 2 ) c) 4xxxx(12yy 3xx) d) 12xxxx(4yy xx) Q5. Fully factorise the area to determine the perimeters for each of these rectangles. a) b) 6

8 Product of Two Brackets Q1. Expand these pairs of brackets and simplify the result: a) (xx + 2)(xx + 4) b) (xx + 5)(xx + 1) c) (xx + 8)(7 + xx) d) (xx 3)(xx + 7) e) (xx 4)(1 + xx) f) (8 xx)(xx + 9) g) (xx + 1)(xx 5) h) (xx + 6)(4 xx) i) (xx + 4)(4 xx) j) (xx 2)(xx 1) k) (xx 6)(xx 4) l) (xx 8)(xx 3) Q2. Expand these pairs of brackets and simplify the result: a) (2xx + 2)(xx + 1) b) (4xx + 2)(xx + 4) c) (5xx + 2)(xx + 3) d) (xx 8)(3xx + 9) e) (xx 5)(5xx + 9) f) (8 + xx)(2xx + 9) g) (4xx + 1)(5 xx) h) (3xx + 1)(6xx 5) i) (1 + 2xx)(5 6xx) j) (2xx aa)(aa 4xx) k) (5xx 2aa)(3aa 6xx) l) (8xx tt)(3xx tt) Q3. Expand these pairs of brackets and simplify the result: a) (xx + 3) 2 b) (5 + xx) 2 c) (xx 9) 2 d) (3xx + 1) 2 e) (4 + 3xx) 2 f) (2aa 4xx) 2 Q4. Write the area of each shape in terms of xx. a) b) c) d) 7

9 Factorising Quadratics when a = 1 Q1. Factorise the following quadratics: a) xx xx + 50 b) xx 2 + 7xx + 6 c) xx xx + 56 d) xx xx + 12 e) xx xx + 40 f) xx 2 8xx + 12 g) xx 2 12xx + 35 h) xx 2 + xx 30 i) xx 2 10xx + 21 j) xx 2 2xx 15 k) xx 2 + xx 20 l) xx 2 + xx 42 m) xx 2 + 7xx + 6 n) xx 2 + 6xx 7 o) xx 2 + 7xx + 6 p) xx 2 5xx + 6 q) xx 2 + 3xx 40 r) xx xx + 99 s) xx 2 6xx + 9 t) xx xx 12 u) xx 2 + 9xx 10 Q2. Expand these pairs of brackets: a) xx 2 64 b) xx 2 9 c) xx 2 4 d) xx 2 49 e) xx f) xx 2 81 g) 16xx 2 81 h) 36xx 2 4 i) 9xx 2 16 Q3. Match the equivalent quadratic expressions: Q4. Write the perimeter of these shapes in terms of xx. 8

10 Factorising Quadratics when a 1 Q1. Fully factorise each of these expressions: a) 2xx xx + 12 b) 3xx xx + 5 c) 2xx xx + 30 d) 3xx xx + 42 e) 3xx 2 + 9xx + 6 f) 5xx xx + 5 g) 2xx 2 + 8xx + 6 h) 2xx 2 + 8xx 24 i) 3xx xx 5 j) 2xx 2 12xx + 16 k) 5xx 2 + 7xx 6 l) 5xx xx 9 m) 2xx 2 xx 6 n) 3xx 2 xx 20 o) 2xx 2 3xx 9 p) 3xx 2 10xx 8 q) 2xx 2 3xx + 1 r) 3xx 2 7xx 20 Q2. Expand these pairs of brackets: a) 4xx 2 20xx + 25 b) 6xx 2 + xx 12 c) 4xx 2 + 3xx 7 d) 6xx ff 63 e) 9xx xx + 8 f) 8xx 2 22xx + 15 Q3. Match together the equivalent expressions: Q4. Write the perimeter of these rectangles in terms of xx. a) b) 9

11 Collecting Like Terms Solutions Q1. a) 4a b) 3rr + 2gg c) 5ee + aa d) 7ff + 4ww e) 5h + 7uu + 5 f) 11xx + 12 g) 5ww + 2bb + 2 h) dd i) 8vv + aa + 2 j) 6 k) 0 l) 9gg + tt + 19 Q2. a) 6aaaa b) 6ffff c) 4aaaaaa d) 6xx 2 e) 4xx 2 + 4rr 2 f) 5dd 2 + 5dd g) 6yy gg 2 h) 5xx 2 i) 5yy 2 + 5yy j) 2aaaa 2 + aaaa 2 + 2bb + 2aa k) 3rr rr + 3 l) 3tt tt + 2jj 2 Q3. Q4. a) Perimeter = 14cc + 8aa + 6 uunniittss b) Perimeter = 7ff + 3 uunniittss 10

12 Algebraic Products Solutions Q1. a) 3aa b) 5rr c) 14rr d) rrrr e) 3eeh f) 16iiii g) 6dd h) 24dd i) 36gg j) 2rrrr k) 8qqqqqq l) 54wwwwww Q2. a) yy 2 b) 2xx 2 c) 8rr 2 d) 2h 2 ff 2 e) 6pppp f) 6xx 2 yy 2 g) 6ii 3 rr h) 16mm 2 nn i) 12gg 2 ffff j) 16yy 2 xx 2 ee k) 16gg 3 ee 2 l) 24rr 3 ff 3 Q3. Q4. a) Possible perimeters include: 8a + 2b, 4a + 4b, 8b + 2a (not exhaustive) b) Possible perimeters include: 72r 2 + 2h, 36r + 4hr 2, 18h + 2h 2, r 2 h, 4h (not exhaustive) 11

13 Substituting into Expressions Solutions Q1. a) 16 b) 7 c) 1 d) 40 e) 36 f) 10 g) 1 h) 4 i) 6 j) 9 k) 14 l) 6 Q2. a) 12 b) 38 c) 16 d) 420 e) 22 f) 65 g) 2 h) 4 i) 1 j) 5 k) 62 l) 634 Q3. 2dd + 6 (46) 4tt + uu (27) 5uu + 6 (7) 3 dd + tt + 1 (27) dd + mm (32) tt (6) uuuu tt (6) 3mm + 10 (46) 6tt 4 (32) uuuu 11 (7) Q4. 4mm + rr (15) 2tt mmmm (18) dddd + 5mm (25) 3tt + rrrr (26) 12

14 Expanding Algebraic Expressions Solutions Q1. a) 3xx + 6 b) 4xx + 24 c) uu d) 7xx 28 e) 4aa + 20 f) 10cc 10 g) 10 2yy h) 8 + 8gg + 8h i) 3ff rr j) 8ee + 20 k) 36 9tt + 18ee l) 42yy tt Q2. a) xx 2 + xx b) ww 2 ww c) 3tt + tt 2 d) 3gg 2 + 2gg e) 2hxx h 2 f) 10dd 12dddd g) 10tt 2 25tttt h) 4jjjj + 8jj i) 8uuuu 40uu 2 j) 8ss 2ss 2 +4us k) 5bb 2 30bbbb + 25bbbb l) 3mmh + 12mmmm + 9mm 2 Q3. a) 5xx + 11 b) 7gg + 27 c) 9aa + 22 d) 8cc + 70 e) 1 3ff f) 16xx + 12 g) 14yy 26 h) 23 21tt i) 21rr 12 Q4. a) AAAAAAAA = 6xx + 26 ssssssssssss uuuuuuuuuu b) AAAAAAAA = 13xx + 18 ssssssssssss uuuuuuuuuu 13

15 Factorising Expressions Solutions Q1. a) 4(h + 3) b) 2(aa + 4) c) 7(tt 2) d) 8(yy 3) e) 6(gg + 2) f) 4(2yy 1) g) 5(uu + 2tt + 3) h) 4(ww 2uu + 3tt) i) 3(2tt + bb 3ii) j) 3(rr + 4 9kk) k) 4(8e 1) l) 7(6jj + rr) Q2. a) aa(cc + bb) b) rr(dd 3) c) gg(yy 3) d) ff(6tt + 5h) e) 6gg(2jj + 3ll) f) 11uu(4h + tt + 3) g) bb(aa + 1) h) 4ff(cc + 2ff) i) ww(ww + 1) j) xx(1 xx) k) 3c(c + 2) l) 5aaaa(bb + 2) Q3. Q4. a) Perimeter = 4a + 12 units b) Perimeter = 72a + 4b units 14

16 Product of Two Brackets Solutions Q1. a) xx 2 + 6xx + 8 b) xx 2 + 6xx + 5 c) xx xx + 56 d) xx 2 + 4xx 21 e) xx 2 3xx 4 f) xx 2 xx + 72 g) xx 2 4xx 5 h) xx 2 2xx + 24 i) 16 xx 2 j) xx 2 3xx + 2 k) xx 2 10xx + 24 l) xx 2 11xx + 24 Q2. a) 2xx 2 + 4xx + 2 b) 4xx xx + 8 c) 5xx xx + 6 d) 3xx 2 15xx 72 e) 5xx 2 16xx 45 f) 2xx xx + 72 g) 4xx xx + 5 h) 18xx 2 9xx 5 i) 12xx 2 + 4xx + 5 j) aa 2 + 6aaaa 8xx 2 k) 6aa aaaa 30xx 2 l) tt 2 11tttt + 24xx 2 Q3. a)xx 2 + 6xx + 9 b) xx xx + 25 c) xx 2 18xx + 81 d) 9xx 2 + 6xx + 1 e) 9xx xx + 16 f) 4aa 2 16aaaa + 16xx 2 Q4 a) xx xx + 28 b) 11xx xx

17 Factorising Quadratics when a = 1 Solutions Q1. a) (xx + 10)(xx + 5) b) (xx + 6)(xx + 1) c) (xx + 7)(xx + 8) d) (xx + 12)(xx + 1) e) (xx + 5)(xx + 8) f) (xx 6)(xx 2) g) (xx 5)(xx 7) h) (xx 5)(xx + 6) i) (xx 3)(xx 7) j) (xx 5)(xx + 3) k) (xx 4)(xx + 5) l) (xx + 7)(xx 6) m) (xx + 6)(xx + 1) n) (xx + 7)(xx 1) o) (xx + 2)(xx + 5) p) (xx 6)(xx + 1) q) (xx + 5)(xx 8) r) (xx + 9)(xx + 11) s) (xx 3)(xx 3) t) (xx + 12)(xx 1) u) (xx + 10)(xx 1) Q2. a) xx ± 8 b) xx ± 3 c) xx ± 2 d) xx ± 7 e) xx ± 13 f) xx ± 9 g) 4xx ± 9 h) 6xx ± 2 i) 3xx ± 4 Q3. Q4. a) Perimeter = 4xx + 26 uuuuuuuuuu b) Perimeter = 4xx 18 uuuuuuuuuu 16

18 Factorising Quadratics when a 1 Solutions Q3. Q4. a) Perimeter = 8xx uuuuuuuuuu b) Perimeter = 10xx 2 uuuuuuuuuu 17