Download Algebra revision exercises(with solutions)

Algebra Revision Exercises
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Question 1
For each of the following pairs of terms, state whether they are like or unlike.
(a) 3x, 4x
(m) 11a4 b5 , −900b4 a5
(b) 5xy, 12xy
(n) 3c2 d3 e, −2d2 ec2
(c) 6, 3x
(o) 14x1 , 3x
(d) 7xy, −2yx
(p) 2xyz, −3yz
(e) 14x2 y 2 , 3yx2
(q) 5, −3x0
(f ) 14, 3
(r) −2yx, 3xyz 0
(g) 6x2 , −x2
(s) 2xyz, 3xtyz
(h) x, −x
(t) 14x, 2xy 0 z 0
(i) 4x2 y 3 z, y 3 zx2
(u) 3xy(z + 2), 2xy(z + 2)
(j) −18pq, 3q
(v) 2xy(z − 3)2 , −xy 2 (z − 3)
(k) 14mn, 11m2 n
(w) 14(x − 2)(y + 3)2 , 60(y + 3)2 (x − 2)
(l) 2rs2 , s2 r
(x) 3x3 (y − z)2 w, 2w(y − z)2 x3
Question 2
Simplify each of these expressions.
(a) 5x + 3x
(e) 3x + 4x + 5x
(i) 2xy − 7xy
(b) 17x + x
(f ) 7x − 5x + 2x
(j) 14x2 y + 2x2 y − yx2
(c) 10x − 2x
(g) 16x − 20x + 2x
(k) 19xyz 2 − 20xz 2 y
(d) 15x − x
(h) 9xy + 12xy
(l) 3x2 y 3 z − 15zx2 y 3
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Algebra Revision Exercises
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Question 3
Simplify each of these expressions.
(a) 5x + 3y + 2x + 8y
(g) 7xyz + 9x − 6y + 3x + 13y − zyx
(b) 14x − 6y + 2x − 3y
(h) x3 + y 3 − x3 − y 3
(c) 4xy + 6x − 2xy − 3x
(i) x + 6 − 4x + 2
(d) 6xy + 14y + yx − 15y
(j) 1 + x + x2 + x3 + x2 + x + 1
(e) 4x2 + 3x2 + 6y − y
(k) 7 − x2 + 4x2 − 4 + 2x2 y
(f ) 2x2 y − 6x + 3yx2 + 4x
(l) 4x2 yz 2 + 6xy 2 z 2 − 15z 2 xy 2 − 19z 2 yx2
Question 4
Calculate the following multiplications of terms.
(a) (3x)(x)
(g) (9r2 )(3r)(r5 )
(b) (5x)(4x)
(h) (6xy)(2x)
(c) (3y 2 )(2y)
(i) (8xy)(2x2 y)
(d) (6x)(2x)(x)
(j) (−xy 2 )(x2 y 2 )
(e) (3x3 )(2x2 )
(k) (−pqr2 )(4qp2 r4 )
(f ) (6r2 )(3r5 )
(l) (−4x2 z 7 y 3 )(−3y 2 xz 2 )(−xyz)
Question 5
Expand each of the following expressions.
(a) 5(x + 2)
(g) q(8 + q)
(b) 6(p + 4)
(h) q 2 (3 + q)
(c) 3(p − 9)
(i) y(7 − y − y 2 )
(d) 2(6 − q)
(j) −y(4y 2 + y)
(e) p(2 + p)
(k) 3y(2y + 7)
(f ) −x(x − 3)
(l) −6y(y − 4)
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Algebra Revision Exercises
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Question 6
Expand each of the following expressions.
(a) 5x(x + 2)
(g) 3pq(2qp2 + q)
(b) 6xy(3 + 2y)
(h) 9a2 b3 (a3 + b2 )
(c) 4xy(x − 3)
(i) 7xy(4 + x2 y − 6y 3 x)
(d) 2pq(3p − 6q)
(j) −2x2 (4 − x2 y − y 3 x)
(e) 3pq(2pq + p)
(k) 6r4 s7 (−3rs + s4 r7 − 2sr2 + r)
(f ) 4p2 (q 2 − 3p2 )
(l) −3z 3 t(−t5 z + t2 z 7 − 8z 3 t9 − zt4 )
Question 7
Expand each of the following expressions. Simplify your answer (if possible).
(a) (x + 1)(x + 3)
(i) (x − 3)(x − 5)
(b) (x + 2)(x + 5)
(j) (−w − 6)(w − 3)
(c) (y + 3)(y + 3)
(k) (r + 9)(r − 9)
(d) (x + 4)(x + 4)
(l) (z + 3)(−z − 3)
(e) (z + 3)(z − 5)
(m) (5 + z)(z + 3)
(f ) (r − 4)(r + 1)
(n) (x + 5)(4 − x)
(g) (x − 7)(x + 1)
(o) (x − 3)(3 − x)
(h) (p − 1)(p − 2)
(p) (5 − x)(−7 − x)
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Algebra Revision Exercises
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Question 8
Expand each of the following expressions. Simplify your answer (if possible).
(a) (2x + 1)(x + 3)
(i) (4 + 2x)(3 + 5y)
(b) (m + 2)(4m + 6)
(j) (6 + 8m)(6 − 2x)
(c) (2m + 2)(3m + 4)
(k) (m2 + 1)(m − 3)
(d) (2m − 1)(3m + 4)
(l) (x2 + 3)(x2 + 5)
(e) (3 − 5m)(m + 6)
(m) (x + y)(z + w)
(f ) (5 − m)(2 − 3m)
(n) (3x2 + 3)(2x2 − 4)
(g) (2m + 3)(−3 − 2m)
(o) (3x3 + y)(2x4 − y 2 )
(h) (3 + x)(2 + y)
(p) (3x2 y − xy 2 )(−3x3 y 2 − yx2 )
Question 9
Factorise each of the following expressions.
(a) 15x + 25
(e) 9x2 y 2 + 3xy
(b) 3x2 − 9x
(f ) x + x2 + x3
(c) 4xy + 40x2
(g) 2x + 3y
(d) 7x2 yz − 8y
(h) 16x2 y 2 − 8x2 y + 9y
Question 10
Factorise each of the following quadratic expressions.
(a) x2 + 3x + 2
(h) x2 + 3x − 10
(b) x2 + 5x + 6
(i) x2 + 4x − 5
(c) x2 + 10x + 21
(j) x2 − 3x + 2
(d) x2 + 8x + 16
(k) x2 − 7x + 10
(e) x2 + 4x + 4
(l) x2 − 5x − 6
(f ) x2 + 9x + 20
(m) x2 − 4x + 4
(g) x2 + 13x + 30
(n) x2 − 11x + 24
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Algebra Revision Exercises
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Question 11
Factorise each of the following quadratic expressions by first factoring out the highest
common factor.
(a) 3x2 + 9x + 6
(h) 11x2 + 33x − 110
(b) 4x2 + 20x + 24
(i) x3 + 3x2 + 2x
(c) 5x2 + 35x + 50
(j) x3 + 5x2 + 6x
(d) 7x2 + 56x + 112
(k) 2x3 + 14x2 + 24x
(e) 2x2 + 16x + 14
(l) 3x4 + 18x3 + 15x2
(f ) 3x2 + 18x + 15
(m) x7 − 4x6 + 4x5
(g) 5x2 + 65x + 150
(n) 4x10 + 20x9 + 24x8
Question 12
Use the difference of two squares identity to factorise each of the following;
(a) x2 − 4
(e) x2 − 100
(i) 49 − x2
(b) x2 − 9
(f ) 4x2 − 9
(j) 9 − x2
(c) x2 − 25
(g) 64x2 − 16
(k) 4 − 25x2
(d) x2 − 1
(h) 81x2 − 36
(l) −16 + 49x2
Question 13
Factorise each of the following by first taking out the highest common factor and then
using the difference of two squares identity.
(a) 3x2 − 27
(e) 3x2 − 300
(i) 50 − 2x2
(b) 2x2 − 18
(f ) 13x2 − 52
(j) 72 − 2x2
(c) 7x2 − 28
(g) 128x2 − 32
(k) 40 − 250x2
(d) 2x2 − 2
(h) 81x2 − 36
(l) −48 + 147x2
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July 2016
Algebra Revision Exercises
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Solutions
Question 1
(a) like
(m) unlike
(b) like
(n) unlike
(c) unlike
(o) like
(d) like
(p) unlike
(e) unlike
(q) like (−3x0 = −3 × 1 = −3)
(f ) like
(r) like
(g) like
(s) unlike
(h) like
(t) like
(i) like
(u) like
(j) unlike
(v) unlike
(k) unlike
(w) like
(l) like
(x) like
Question 2
(a) 8x
(e) 12x
(i) −5xy
(b) 18x
(f ) 4x
(j) 15x2 y
(c) 8x
(g) −2x
(k) −xyz 2
(d) 14x
(h) 21xy
(l) −12x2 y 3 z
Question 3
(a) 7x + 11y
(e) 7x2 + 5y
(i) −3x + 8
(b) 16x − 9y
(f ) 5x2 y − 2x
(j) 2 + 2x + 2x2 + x3
(c) 2xy + 3x
(g) 6xyz + 12x + 7y
(k) 3 + 3x2 + 2x2 y
(d) 7xy − y
(h) 0
(l) −15x2 yz 2 − 9xy 2 z 2
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July 2016
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Question 4
(a) 3x2
(e) 6x5
(i) 16x3 y 2
(b) 20x2
(f ) 18r7
(j) −x3 y 4
(c) 6y 3
(g) 27r8
(k) −4p3 q 2 r6
(d) 12x3
(h) 12x2 y
(l) −12x4 y 6 z 10
Question 5
(a) 5x + 10
(e) 2p + p2
(i) 7y − y 2 − y 3
(b) 6p + 24
(f ) −x2 + 3x
(j) −4y 3 − y 2
(c) 3p − 27
(g) 8q + q 2
(k) 6y 2 + 21y
(d) 12 − 2q
(h) 3q 2 + q 3
(l) −6y 2 + 24y
Question 6
(a) 5x2 + 10x
(g) 6q 2 p3 + 3pq 2
(b) 18xy + 12xy 2
(h) 9a5 b3 + 9a2 b5
(c) 4x2 y − 12xy
(i) 28xy + 7x3 y 2 − 42x2 y 4
(d) 6p2 q − 12pq 2
(j) −8x2 + 2x4 y + 2x3 y 3
(e) 6p2 q 2 + 3p2 q
(k) −18r5 s8 + 6r11 s11 − 12r6 s8 + 6r5 s7
(f ) 4p2 q 2 − 12p4
(l) 3t6 z 4 − 3t3 z 10 + 24t10 z 6 + 3t5 z 4
Question 7
(a) x2 + 4x + 3
(g) x2 − 6x − 7
(m) z 2 + 8z + 15
(b) x2 + 7x + 10
(h) p2 − 3p + 2
(n) −x2 − x + 20
(c) y 2 + 6y + 9
(i) x2 − 8x + 15
(o) −x2 + 6x − 9
(d) x2 + 8x + 16
(j) (−w2 − 3w + 18
(p) x2 + 2x − 35
(e) z 2 − 2z − 15
(k) r2 − 81
(f ) r2 − 3r − 4
(l) −z 2 − 6z − 9
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July 2016
Algebra Revision Exercises
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Question 8
(a) 2x2 + 7x + 3
(i) 10xy + 6x + 20y + 12
(b) 4m2 + 14m + 12
(j) −16mx + 48m − 12x + 36
(c) 6m2 + 14m + 8
(k) m3 − 3m2 + m − 3
(d) 6m2 + 5m − 4
(l) x4 + 8x2 + 15
(e) −5m2 − 27m + 18
(m) xz + xw + yz + wy
(f ) 3m2 − 17m + 10
(n) 6x4 − 6x2 − 12
(g) −4m2 − 12m − 9
(o) 6x7 − 3x3 y 2 + 2x4 y − y 3
(h) xy + 2x + 3y + 6
(p) −9x5 y 3 − 3x4 y 3 + 3x4 y 4 + x3 y 3
Question 9
(a) 5(3x + 5)
(d) y(7x2 z − 8)
(g) cannot be factorised
(b) 3x(x − 3)
(e) 3xy(3xy + 1)
(h) y(16x2 y − 8x2 + 9)
(c) 4x(y + 10x)
(f ) x(1 + x + x2 )
Question 10
(a) (x + 1)(x + 2)
(f ) (x + 4)(x + 5)
(k) (x − 5)(x − 2)
(b) (x + 2)(x + 3)
(g) (x + 10)(x + 3)
(l) (x − 6)(x + 1)
(c) (x + 7)(x + 3)
(h) (x + 5)(x − 2)
(m) (x − 2)2
(d) (x + 4)2
(i) (x + 5)(x − 1)
(n) (x − 8)(x − 3)
(e) (x + 2)2
(j) (x − 2)(x − 1)
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July 2016
Algebra Revision Exercises
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Question 11
(a) 3(x + 1)(x + 2)
(f ) 3(x + 1)(x + 5)
(k) 2x(x + 3)(x + 4)
(b) 4(x + 2)(x + 3)
(g) 5(x + 10)(x + 3)
(l) 3x2 2(x + 5)(x + 1)
(c) 5(x + 2)(x + 5)
(h) 11(x + 5)(x − 2)
(m) x5 (x − 2)2
(d) 7(x + 4)2
(i) x(x + 1)(x + 2)
(n) 4x8 (x + 2)(x + 3)
(e) 2(x + 1)(x + 7)
(j) x(x + 2)(x + 3)
Question 12
(a) (x − 2)(x + 2)
(g) (8x − 4)(8x + 4) = 4(2x − 1)(2x + 1)
(b) (x − 3)(x + 3)
(h) (9x − 6)(9x + 6) = 9(x − 2)(x + 2)
(c) (x − 5)(x + 5)
(i) (7 − x)(7 + x)
(d) (x − 1)(x + 1)
(j) (3 − x)(3 + x)
(e) (x − 10)(x + 10)
(k) (2 − 5x)(2 + 5x)
(f ) (2x − 3)(2x + 3)
(l) (7x − 4)(7x + 4)
Question 13
(a) 3(x − 3)(x + 3)
(e) 3(x − 10)(x + 10)
(i) 2(5 − x)(5 + x)
(b) 2(x − 3)(x + 3)
(f ) 13(x − 2)(x + 2)
(j) 2(6 − x)(6 + x)
(c) 7(x − 2)(x + 2)
(g) 32(2x − 1)(2x + 1)
(k) 10(2 − 5x)(2 + 5x)
(d) 2(x − 1)(x + 1)
(h) 9(3x − 2)(3x + 2)
(l) 3(7x − 4)(7x + 4)
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