Basic Arithmetic & Algebra (1.1 – 1.4)

 

1.1              Review of arithmetical operations on rational numbers and quadratic surds.

1.2              Inequalities and absolute values.

1.3              Review of manipulation of and substitution in algebraic expressions, factorisation, and operations on simple algebraic fractions.

1.4              Linear equations and inequalities. Quadratic equations. Simultaneous equations.

 

 

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BASIC ARITHMETIC I

 

Without using your calculator, evaluate the following:

 

Q.1.     (a) 6 - 3                        (b) 3  + 4             (c) 5   6        

(d) 2  3          

 

Q.2.     (a) (2½)3                      (b) (0.4)2          (c) (1.2)2          (d) (0.2)3

 

Q.3.     Write the following as fractions in their simplest form:

            (a) 0.6              (b) 0.125          (c) 40%            (d) 65%

 

Q.4.     Write the following as decimals:

            (a)             (b)                (c) 85%            (d) 7 ½ %

 

Q.5. Evaluate the following to 3 decimal places:

            (a)                (b) 4 ½ %        (c) 1            (d) 12 ½ %

 

Q.6. If v2 = u2 + 2as calculate the value of s when v = 5, u = 3, a = 4

 

Q.7. Find the mean of the following numbers:

            (a) 7,4,3,6,0.    (b) 6,8,6,4.       (c) 15, 12, 13, 13,17

            (d) 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124

 

Q.8. Write the following as ordinary numbers:

            (a) 2.4 x 102     (b) 6.46 x 103   (c) 4.6 x 10-2    (d) 3.273 x 10-1

 

Q.9. A motorist leaves Nathantown at 10.00 a.m. and arrives at Joshville, 210 km away at 1.30 p.m. What is the average speed of the motorist?

 

Q.10.   Find the square root of the following numbers:

            (a) 121             (b) 0.04                        (c) 106              (d) 6.4 x 105

 

ANSWERS:

Q.1. (a) 2   (b) 7             (c) 34   (d)

Q.2. (a) 125/8 or 15 5/8           (b) 0.16            (c) 1.44            (d) 0.008

Q.3. (a)        (b)                (c)                (d)

Q.4. (a) 0.32    (b) 0.375          (c) 0.85            (d) 0.075

Q.5. (a) 0.143  (b) 0.045          (c) 1.182          (d) 0.125          Q.6. s = 2

Q.7. (a) 4         (b) 6                 (c) 14               (d) 118

Q.8. (a) 240     (b) 6460           (c) 0.046          (d) 0.3273        Q.9. 60 km/h

Q.10. (a) 11     (b) 0.2              (c) 103 or 1000            (d) 800

 

BASIC ARITHMETIC II

Without using your calculator, evaluate the following:

 

Q.1.     (a)       (b)       (c)       (d)    

 

Q.2.     (a)         (b) (0.3)3          (c) (1.3)2          (d) (0.2)4

 

Q.3.     Write the following as fractions in their simplest form:

            (a) 0.8              (b) 0.375          (c) 60%            (d) 85%

 

Q.4.     Write the following as decimals:

            (a)              (b)                (c) 63%            (d) 12 ½ %

 

Q.5. Evaluate the following to 3 decimal places:

            (a)                (b) 6 ½ %        (c)              (d) 14 ½ %

 

Q.6. If s = ut + ½ at2 calculate the value of a when s = 80, u = 5, t = 2

 

Q.7. Find the mean of the following numbers:

            (a) 8,5,5,0,2.    (b) 6,1,12,5.     (c) 21, 29, 13, 18,19

            (d) 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240

 

Q.8. Write the following as ordinary numbers:

            (a) 8.3 x 103     (b) 6.462 x 102             (c) 3.2 x 10-2    (d) 46.874 x 10-1

 

Q.9. A motorist leaves Amytown at 9.30 a.m. and arrives at Stephhville, 375 km away at 2.30 p.m. What is the average speed of the motorist?

 

Q.10.   Find the square root of the following numbers:

            (a) 169             (b) 0.01                        (c) 108              (d) 8.1 x 107

 

ANSWERS:

Q.1. (a)     (b)  (c) 17          (d)

Q.2. (a)  or     (b) 0.027          (c) 1.69            (d) 0.0016

Q.3. (a)        (b)                (c)                (d)

Q.4. (a) 0.34    (b) 0.625          (c) 0.63            (d) 0.125

Q.5. (a) 0.333  (not ) (b) 0.065      (c) 1.429          (d) 0.145          Q.6. a = 35

Q.7. (a) 4         (b) 6                 (c) 20               (d) 234

Q.8. (a) 8300   (b) 646.2          (c) 0.032          (d) 4.6874        Q.9. 75 km/h

Q.10. (a) 13     (b) 0.1              (c) 104 or 10000          (d) 9000

BASIC ALGEBRA I

 

Q.1.     Express the following in their simplest form:

            (a) 2a + 3b + 6a – 8b   (b) 2mn + 3pq – 3mn – pq       (c) 11x + 2y –z – 2y + x

            (d) x2 + y2 + 3xy - 4 + 2xy + 8 (e) 2cd – 2ef + 5dc – 2fe + cd2

 

Q.2.     Simplify:           (a) g3 x g5         (b) m6 χ m4      (c) (k2)4            (d) p2 χ p5

            (e) (2g)3           (f) (pq)2            (g) pq4 χ q2      (h) (k2)-2           (i) r3s2 x (rs)2

 

Q.3.     Simplify the following:  (a) (b)        

(c)             (d)       (e)    

(f)          (g)         (h)

Q.4.     Expand the following:

            (a) (2x – y)(2x + y)       (b) (3b – a)2    (c) (x + 5)(x – 3)   (d) 2b(x – 4) -6(x – 4)

            (e) (2x – 4)(3x + 6)      (f) (x – 3)(x2 + 3x + 9)  (g) (2xy + z)(4xy – z)

 

Q.5.     Factorise the following:

            (a) (a2 – 4)     (b) (4m2 – 9n2)     (c) k2 + 2k + 1     (d) 4g2 – 12gh + 9h2

            (e) 8p3 + q3      (f) p2 + 9p + 14   (g) k2 – 3k – 18   (h) 2g2 + 9g – 18

 

Q.6.     Solve the following equations:

            (a) 5x = 15   (b) 2x + 4 = 12     (c) 4g – 6 = 2g + 4     (d) 3m + 4 = m – 8

            (e) 2k – 3 = 7k + 7     (f) 2 – p = 6 – 2p     (g) 6q + 18 + 3q = 3 – q

 

Q.7.     An apricot costs 10 cents more than a peach. If 2 peaches plus 3 apricots cost $2.80, what is the cost of a peach and what is the cost of an apricot?

Answers:

Q.1.     (a) 6a – 5b       (b) 2pq – mn    (c) 12x – z        (d) x2 + y2 + 5xy + 4

            (e) 7cd – 4ef + cd2

Q.2.     (a) g8    (b) m2   (c) k8    (d) p-3 or    (e) 8g3    (f) p2q2    (g) pq2   (h) k-4 or    (i) r5s4

Q.3.     (a)    (b)    (c)    (d)    (e)    (f)    (g) ab5   (h)

Q.4.     (a) 4x2 – y2       (b) 9b2 – 6ab + a2    (c) x2 + 2x – 15    (d) 2bx – 8b – 6x + 24

            (e) 6x2 – 24      (f) x3 – 27         (g) 8x2y2 + 2xyz – z2

Q.5.     (a) (a + 2)(a – 2)     (b) (2m + 3n)(2m – 3n)     (c) (k + 1)2     (d) (2g – 3h)2

            (e) (2p + q)(4p2 – 2pq + q2)     (f) (p + 7)( p + 2)     (g) (k – 6)(k + 3)

            (h) (2g – 3)(g + 6)

Q.6.     (a) x = 3   (b) x = 4    (c) g = 5    (d) m = -6    (e) k = -2    (f) p = 4   (g) q = -1½

Q.7.     Let cost of peach be x. cost of apricot = (x + 10)

            2x + 3(x + 10) = 280               2x + 3x + 30 = 280      5x = 250          x = 50

            peach = 50 cents, apricot = 50 + 10 = 60 cents

 

BASIC ALGEBRA II

 

Q.1.     Express the following in their simplest form:

            (a) 5d + 16e – 6d + 4e     (b) 4ab + 6cd – 2ab – 3c     (c) 2p + 3q – r – 2q + 2r 

            (d) x2 + y2 - 4xy + 6 + 2xy + 3 (e) 5pq – 2rs – 3qp – 2sr + pq + 4rs

 

Q.2.     Simplify:           (a) p4 x p7         (b) r8 χ r5         (c) (b4)3            (d) g3 χ g7

            (e) (4d)2           (f) (2ab)3          (g) x2y4 χ y3      (h) (m3)-2          (i) (mn)3 χ m2n

 

Q.3.     Simplify the following:  (a)    (b)         

(c)             (d)        (e)    

(f)          (g)       (h)

Q.4.     Expand the following:

            (a) (m + 3n)(m – 3n)     (b) (g + 2h)2    (c) (x + 4)(x – 5)   (d) 3k(j + 3) -4(j + 3)

            (e) (4p - 2)(3p + 4)      (f) (r + 2)(r2 – 2r + 4)   (g) (3pq + r)(2pq – r)

 

Q.5.     Factorise the following:

            (a) (4p2 – 9)     (b) (2g2 – 18)     (c) m2 – 2m + 1     (d) 16r2 + 24rs + 9s2

            (e) g3 – 27h3     (f) m2 - m - 12   (g) 2q2 + 3q – 20   (h) 3n2 - 2n – 21

 

Q.6.     Solve the following equations:

            (a) 6x = 24   (b) 3x + 2 = 4x - 6     (c) 7g – 5 = 2g + 10     (d) 5k + 9 = 23 – 2k

            (e) 2m + 6 + m = 5m - 4     (f) 5 – g + 10 – 3g  = 12 + 4g - 5    

(g) 2q + 3  = 7q + 8  – q + 5

 

Q.7.     A custard tart costs 80 cents more than a lamington. If 2 lamingtons plus a custard tart cost $3.20, what is the cost of a custard tart, and what is the cost of a lamington?

 

Answers:

Q.1.     (a) 20e - d        (b) 2ab – 3cd   (c) 2p + q + r   (d) x2 + y2 - 2xy + 9   (e) 3pq

Q.2.     (a) p11  (b) r3    (c) b12  (d) g-4 or    (e) 16d2    (f) 8a3b3    (g) x2y   (h) m-6 or    (i) mn2

Q.3.     (a)  (b) gjk   (c)   (d)    (e)   (f)   (g) f 2 e3   (h)

Q.4.     (a) m2 – 9n2      (b) g2 + 4gh + 4h2    (c) x2 - x – 20    (d) 3jk – 4j + 9k - 12

            (e) 12p2 + 10p - 8        (f) r3 + 8           (g) 6p2q2 - rpq – r2

Q.5.     (a) (2p + 3)(2p – 3)     (b) 2(g + 3)(g – 3)     (c) (m - 1)2     (d) (4g + 3s)2

            (e) (g - 3h)(g2 + 3gh + 9h2)     (f) (m - 4)( m + 3)     (g) (q + 4)(2q - 5)

            (h) (3n + 7)(n - 3)

Q.6.     (a) x = 4   (b) x = 8    (c) g = 3    (d) k = 2    (e) m = 5   (f) g = -1   (g) q = -2½

Q.7.     Lamington = 80 cents, Custard tart = $1.60

FACTORISATION

 

 1.        Multiply the following and express them in their simplest form:

(i) (a + b) (a – b)                                  (ii) (2x + 3y) (2x – 3y)

            (iii) (a + b)2                                           (iv) (m – n)2     

            (v) (a + b) (a2 – ab + b2)                       (vi) (g – h)  (g2 + gh + h2)         

(vii) (3j – 2k) ( 9j2 + 6jk + 4k2)            (viii) (x – 3) (x + 2)

(ix) (L + 5) (L + 7)                               (x) (2d – 4) (3d – 5)

 

2.                  Factorise the following:

(i) p2 – q2         (ii) r2 – 9           (iii) w2 – t2        (iv) s2 – 36       (v) 25 – k2

(vi)16 – 4g2      (vii) 9m2 – 36n2                        (viii) 49p2 – 4q2 

(ix) 8b2 – 18c2              (x) 50y2 – 32z2

 

3.                  Factorise the following:

(i) a3 – b3          (ii) m3 + n3        (iii) g3 + 8         (iv) w3 – 27      (v) 64 – p3

(vi) 8y3 – z3      (vii) 125g3 + 8 (viii) 27p3 – 8q3

(ix) 64r3 + 125s3                       (x) 54m3 + 16n3

 

4.                  Factorise the following:

(i) x2 + x -12    (ii) x2 + 7x + 10            (iii) x2 – 6x – 16            (iv) x2 + 12x + 32

(v) x2 – 10x + 21          (vi) x2 – 10x + 16         (vii) x2 –x – 56

(viii) x2 – 9x + 18          (ix) x2 + 11x + 24         (x) x2 – x - 42

(xi) x2 + 5x – 24           (xii) x2 –x - 90

 

  1. Factorise the following:

(i) 3x2 + 11x + 10         (ii) 4x2 + 5x – 6            (iii) 5x2 + 33x + 18

(iv) 5x2 – 11x – 12       (v) 7x2 – 30x +8           (vi) 5x2 + 17x – 12

(vii) 2x2 + 5x – 42        (viii) 4x2 – 13x - 35      (ix) 7x2 –x -6

(x) 5x2 + 31x – 28        (xi) 6x2 + 5x + 1           (xi) 12x2 + 5x - 2

 

ANSWERS.

 

1.         (i) a2 – b2   (ii) 4x2 – 9y2  (iii) a2 + 2ab + b2   (iv) m2 – 2mn + n2

            (v) a3 + b3  (vi) g3 – h3     (vii) 27j3 – 8k3  (viii) x2 –x – 6  (ix) L2 + 12L + 35

            (x) 6d2 – 22d + 20

2.         (i) (p + q)( p – q)   (ii) (r + 3)( r – 3)   (iii) (w + t)( w – t)   (iv) (s + 6)( s – 6)   (v) (5 + k)( 5 – k)   (vi) (4 + 2g)( 4 – 2g)   (vii) (3m + 6n)( 3m – 6n)  

(viii) (7p + 2q)(7 p – 2q)   (ix) 2(2b + 3)( 2b – 3c)   (x) 2(5y + 4z)( 5y – 4z)  

3.         (i) (a – b)( a2 + ab + b2)   (ii) (m + n)( m2 – mn + n2)  (iii) (g + 2)( g2 – 2g + 4)

            (iv) (w – 3)(w2 + 3w + 9) (v) (4 – p)(16 + 4p +p2)  (vi) (2y – z)(4y2 + 2yz + z2)

            (vii) (5g + 2)(25g2 – 10g +4)      (viii) (3p – 2q)( 9p2 +6pq + 4q2)  

            (ix) (4r + 5s)( 16r2 – 20rs + 25s2)   (x) 2(3m + 2n)( 9m2 – 6mn + 4n2)

4.         (i) (x-3)(x+4)   (ii) (x + 2)( x + 5)          (iii) (x – 8)( x + 2)  (iv) ((x +4)( x + 8)

            (v) (x – 7)( x – 3)  (vi) (x – 8)( x – 2)   (vii) (x + 7)( x – 8)  (viii) (x – 3)( x – 6)

            (ix) (x + 8) (x + 3)  (x) (x + 6) (x – 7)  (xi) (x + 8)( x – 3) (xii) (x + 9)(x – 10)

5.         (i) (x + 2)( 3x + 5)   (ii) (x + 2)( 4x – 3)   (iii) (x + 6)( 5x + 3)

(iv) (5x + 4)( x – 3) (v) (7x – 2)( x – 4) (vi) (5x – 3)(x + 4)

(vii) (2x – 7)( x + 6) (viii) (4x + 7)( x – 5) (ix) (7x + 6)(x – 1)

(x) (5x – 4)( x + 7)   (xi) (2x + 1)(3x + 1)  (xii) (3x + 2)( 4x – 1)

FACTORISING TRINOMIALS

 

 Trinomials are expressions involving three terms. Those of the form ax2 + bx + c, are called “Quadratic trinomials”.

The easiest trinomials to factorise are those where the coefficient of x is 1,

i.e. x2 + bx + c

 

Consider the trinomial x2 + 5x + 6

We need two numbers that multiply to give 6 and add to give 5.

You will recognise these numbers as 2 & 3.

Hence the factors of  x2 + 5x + 6 are (x + 2)(x + 3)

 

Consider the trinomial x2 - 5x + 6

We need two numbers that multiply to give 6 and add to give -5.

You will remember that two negative numbers multiply to give a positive.

So the numbers we want are -2 & -3.

Hence the factors of  x2 - 5x + 6 are (x - 2)(x - 3)

 

Consider the trinomial x2 + x - 6

We need two numbers that multiply to give -6 and add to give +1.

You will remember that a positive and a negative multiply to give a negative.

You will recognise these numbers as -2 & 3.

Hence the factors of  x2 + x - 6 are (x - 2)(x + 3)

 

Consider the trinomial x2 - x - 6

We need two numbers that multiply to give -6 and add to give -1.

You will recognise these numbers as 2 & -3.

Hence the factors of  x2 - x - 6 are (x + 2)(x - 3)

 

So to factorise a trinomial of the form x2 + bx + c

Look at the sign of the c. If it is positive then both factors have the same sign. If it is negative then the factors have different signs.

Look at the sign of the b. If the factors have the same sign, then the sign of the b is the sign of the factors. If the factors have different signs, then the sign of the b is the sign of the larger factor.

Apply these rules to the four examples above.

 

Exercise: Factorise the following:

(i) x2 + 7x + 12 (ii) x2 - 7x + 12        (iii) x2 - x – 12           (iv) x2 + x – 12

(v) x2 + 3x – 4              (vi) x2 - 3x – 4           (vii) x2 - 7x + 10    (viii) x2 + 3x – 10

(ix) x2 - 6x + 8              (x) x2 - 5x – 14         (xi) x2 + 6x + 8        (xii) x2 - x – 20

(xiii) x2 + 4x + 21         (xiv) x2 + x – 30       (xv) x2 + 8x + 12    (xvi) x2 - 2x - 35

(xvii) x2 - 2x - 15          (xiv) x2 - 6x – 16      (xv) x2 - 9x + 20     (xvi) x2 + 11x + 28

 

Answers:

(i) (x + 4)(x + 3)           (ii) (x - 4)(x - 3)            (iii) (x - 4)(x + 3)     (iv) (x + 4)(x - 3)

(v) (x + 4)(x - 1)           (vi) (x - 4)(x + 1)          (vii) (x - 2)(x - 5)     (viii) (x - 2)(x + 5)

(ix) (x - 4)(x - 2)           (x) (x - 7)(x + 2)           (xi) (x + 4)(x + 2)     (xii) (x + 4)(x - 5)

(xiii) (x - 3)(x + 7)        (xiv) (x - 5)(x + 6)        (xv) (x + 6)(x + 2)     (xvi) (x + 5)(x - 7)

(xvii) (x - 5)(x + 3)       (xviii) (x - 8)(x + 2)       (xix) (x - 4)(x - 5)     (xx) (x + 4)(x + 7)

FACTORISATION: COMMON FACTORS

 

The simplest method of factorisation is to take out a common factor

e.g. 2x + 4 = 2(x + 2)

Note that when we multiply everything inside the bracket by the factor we have taken out, we again get the original expression.

 

Examples:

(i) 5x2 + 10x + 20 = 5(x2 + 2x + 4)

(ii) 10x2 + 5x = 5x(2x + 1)

 

Exercise 1: Factorise the following by taking out a common factor.

(i) 5x + 10        (ii) 6g + 3         (iii) x2 + 2x       (iv) 21p2 + 7p + 14      (v) 2k2 + 4k

(vi) 8y2 + 12y + 8         (vii) 5z – 15z2               (viii) 21f 2 – 14 (ix) mn + 5m

(x) m2n – 5m                (xi) m2n + mn2              (xii) p3k2 – pk2             

 

Answers:

Exercise 1: (i) 5(x + 2)     (ii) 3(g + 1)     (iii) x(x + 2)     (iv) 7(3p2 + p + 2)

(v) 2k(k + 2)     (vi) 4(2y2 + 3y + 2)     (vii) 5z(1 – 3z)     (viii) 7(3f 2 – 2)    

(ix) m(n + 5)     (x) m(mn – 5)     (xi) mn(m + n)     (xii) pk2 (p2 – 1)

 

GROUPING in PAIRS

Where there are 4 terms it is often possible to group in pairs so that there is a common factor.

Examples:

(i) ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)

(ii) ax – ay – bx + by = a(x + y) – b(x + y) = (a – b)(x + y)

(iii) 2xp + 3p + 8x + 12 = 2p(x + 3) + 4(x + 3) = (2p + 4)(x + 3)

 

Exercise 2: Factorise the following expressions by grouping in pairs:

(i) mp + np + mq + nq     (ii) 2gh + 8g + hk + 4k          (iii) x3 – 4x2 + 2x – 8

(iv) pq + q2 – p – q            (v) 2pr – 2qr + ps – qs        (vi) 2g2 – 4gh – 3g + 6h

(vii) 2p2 – 6pq – qp + 3q2     (viii) a3 – 3a2 + 6a – 18     (ix) x3 + x2 + x + 1

(x) 15x2 + 3xyz – 10xyz + 2y2z2     (xi) 3pq – 2pr – 4r + 6q     (xii) 3pq – qr – 3p + r

(xiii) m2n2 + 3m2n + 4n + 12    (xiv) 2rs – 8pr – qs + 4pq   (xv) 5f 2g2 + 20fg2 – 3f - 12

(xvi) xyz – yz2 – 4x + 4z      (xvii) 9 – 15z + 27z2 – 45z3     (xviii) x5 – x4 + x3 – x2

(xix) a3b3 + a3b2 – ab3c – ab2c              (xx) g3h – 2g3 – 4g2h – 8g3

 

Answers:

Exercise 2: (i) (m + n)(p + q)     (ii) (2g + k)(h + 4)     (iii) (x2 + 2)(x – 4)

(iv) (p + q)(q – 1)     (v) (2r + s)(p – q)     (vi) (2g – 3)(g – 2h)     (vii) (2p – q)(p – 3q)

(viii) (a2 + 6)(a – 3)     (ix) (x2 + 1)(x + 1)     (x) (5x + yz)(3x – 2yz)

(xi) (p + 2)(3q – 2r)      (xii) (q – 1)(3p – r)     (xiii) (m2n + 4)(n + 3)    

(xiv) (2r – q)(s – 4p)    (xv) (5fg2 – 3)(f + 4)     (xvi) (yz – 4)( x-z)

(xvii) 3(1 + 3z2)(3 – 5z)     (xviii) x2(x2 + 1)(x – 1)     (xix) b2(a3 – ac)(b + 1)

(xx) g(g2 – 4g)(h + 2)

 

 

 

 

FACTORISING TRINOMIALS

 The Cross Method:

This method involves writing the factors of the first term of the trinomial on the left hand side of the cross and the factors of the last term of the trinomial on the right hand side of the cross. The factors are multiplied along the diagonals of the cross and their sum written down. A sum equal to the middle term of the trinomial indicates the correct factors.

Since there are several combinations, it is often a case of trial and error.

Consider the trinomial expression 4x2 + 8x + 3

The possible combinations are:

                       

The third combination gives the correct sum for the middle term.

Collect the terms at the top and the terms at the bottom of the third combination.

                                   

Hence the factors of 4x2 + 8x + 3 are (2x + 1)(2x + 3)

 

Product & Factor Method:

Multiply the first and the last term: 4x2 . 3 = 12x2

Determine the factors of 12x2 that add to 8x.

Possible factors:            12x, x   sum = 13x

                                    6x, 2x   sum = 8x

                                    3x, 4x   sum = 7x

As can be seen from above, the correct factors are 6x and 2x. Insert these instead of 8x.

4x2 + 6x + 2x +3 Factorise by grouping in pairs

2x(2x + 3) + 1(2x + 3)

(2x + 1)(2x + 3)

 

Exercise:

 Factorise the following trinomials by each of the above methods:

(i) 2x2 + 5x + 3     (ii) 2x2 + 7x + 3     (iii) 4x2 + 7x + 3     (iv) 4x2 + 12x + 5

(v) 6x2 + 11x + 6  (vi) 6x2 + 17x + 5  (vii) 6x2 + 31x + 5  (viii) 6x2 + 25x + 25

(ix) 2x2 + 9x + 7   (x) 4x2 + 16x + 7   (xi) 4x2 + 29x + 7    (xii) 8x2 + 34x + 21

 

Answers:

(i) (2x + 3)(x + 1)     (ii) (2x + 1)(x + 3)     (iii) (4x + 3)(x + 1)   (iv) (2x + 5)(2x + 1)

(v) (3x + 2)(2x + 3)  (vi) (2x + 5)(3x + 1)   (vii) (x + 5)(6x + 1) (viii) (2x + 5)(3x + 5)    

(ix) (2x + 7)(x + 1)   (x) (2x + 1)(2x + 7)    (xi) (x + 7)(4x + 1)   (xii) (4x + 3)(2x + 7)    

 FACTORISING TRINOMIALS (negatives)

 The Cross Method:

A cross is drawn and the factors of the first and last terms are written on opposite sides of the cross, being careful to insert correct signs.

There are several combinations, so it is a case of trial and error.

Consider the trinomial expression 6x2 + 5x - 6

The possible combinations are:

                       

                       

The last combination gives the correct sum for the middle term.

Collect the terms at the top and the terms at the bottom of the last combination.

                                   

Hence the factors of 6x2 + 5x - 6 are (3x - 2)(2x + 3)

 

Product & Factor Method:

Multiply the first and the last term: 6x2 .(-6)  = -36x2

Determine the factors of -36x2 that add to 5x.

Possible factors:            (-36x)(x)          sum = -35x

                                    (36x)(-x)          sum = 35x

                                    (-6x)(6x)          sum = 0

(12x)(-3x)        sum = 9x

(-12x)(3x)        sum = -9x

(-9x)(4x)          sum = -5x

(9x)(-4x)          sum = 5x

                                   

As can be seen from above, the correct factors are 9x and -4x. Insert these instead of 5x.

6x2 + 9x - 4x - 6 Factorise by grouping in pairs

3x(2x + 3) -2 (2x + 3)

(3x - 2)(2x + 3)

 

Exercise:

 Factorise the following trinomials by each of the above methods:

(i) 2x2 + 5x - 3     (ii) 2x2 - 9x - 5     (iii) 12x2 - 16x - 3     (iv) 4x2 - 5x - 6

(v) 6x2 - 17x - 3  (vi) 4x2 - 10x + 4  (vii) 15x2 + 17x - 4  (viii) 15x2 + 2x - 8

(ix) 20x2 + x - 12   (x) 6x2 – 17x + 12   (xi)14x2 + 13x - 12    (xii) 21x2 + 2x - 8

 

Answers:

(i) (2x - 1)(x + 3)     (ii) (2x + 1)(x - 5)     (iii) (2x - 3)(6x + 1)   (iv) (4x + 3)(x - 2)

(v) (6x + 1)(x - 3)  (vi) 2(2x - 1)(x - 2)   (vii) (3x + 4)(5x - 1) (viii) (5x + 4)(3x - 2)    

(ix) (5x + 4)(4x - 3)   (x) (3x - 4)(2x - 3)    (xi) (2x + 3)(7x - 4)   (xii) (3x + 2)(7x - 4)    

DIFFERENCE OF SQUARES

 

Multiply the factors (a + b)(a – b)

You should have obtained the result (a + b)(a – b) = a2 + ab – ab – b2 = a2 – b2

This leads to the important result in factorisation that is known as “The difference of squares” i.e. a2 – b2  = (a + b)(a – b)

Example: Factorise 4x2 – 9y2

Answer: 4x2 – 9y2 = (2x)2 – (3y)2 = (2x + 3y)(2x – 3y)

 

Exercise: Factorise the following:

(i) m2 – n2     (ii) p2 – q2     (iii) 9f 2 – g2     (iv) 16k2 – 9h2     (v) 4x2 – 25y2

(vi) 121v2 – 49w2     (vii) 81g2 – 169f 2     (viii) 18x2 – 32y2     (ix) 27p2 – 12q2

(x) 8z2 – 98w2     (xi) 16 k2 – 36 h2     (xii) 100n2 – 198m2

 

Answers:

(i) (m + n)(m – n)     (ii) (p + q)(p – q)     (iii) (3f + g)(3f – g)    (iv) (4k + 3h)(4k – 3h)    

(v) (2x + 5y)(2x – 5y)     (vi) (11v + 7w)( 11v – 7w)     (vii) (9g + 13f)(9g – 13f)

(viii) 2(3x + 4y)( 3x – 4y)     (ix) 3(3p + 2q)(3p – 2q)     (x) 2(2z + 7w)(2z – 7w)

(xi) 4(2k + 3h)(2k – 3h)     (xii) 4(5n + 7m)(5n – 7m)

 

SUM & DIFFERENCE OF CUBES

 

Multiply the factors (a + b)( a2 – ab + b2)

You should get a3 – a2b + ab2 + a2b – ab2 + b3 = a3 + b3

This leads to the important result regarding factorising the sum of two cubes:

 a3 + b3 = (a + b)(a2 – ab + b2)

  

 Multiply the factors (a - b)( a2 + ab + b2)

You should get a3 + a2b - ab2 + a2b – ab2 + b3 = a3 - b3

This leads to the important result regarding factorising the difference of two cubes:

 a3 - b3 = (a - b)(a2 + ab + b2)

 

Example: Factorise 8x3 – y3

Answer: 8x3 – y3 = (2x)3 – y3 = (2x – y)( 4x2 + 2xy + y2)]

 

Exercise:

(i) m3 – n3       (ii) p3 + r3      (iii) a3 + 27      (iv) 27k3 – h3     (v) 8a3 + 27b3   

(vi) 64g3 –125     (vii) 8p3 – 27q3     (viii) 125r3 – 27s3     (ix) 2m3 + 16n3

(x) 81a3 – 3b3   (xi) 40k3 – 5h3 (xii) 24x3 + 81y3     (xiii) 135c3 + 40d3

(xiv) 2ab3c3 + 2ad3       (xv) 3wx3y3 – 24wz3

 

Answers:

(i) (m – n)(m2 + mn + n2)    (ii) (p + r)(p2 – pr + r2)     (iii) (a + 3)( a2 – 3a + 9)

(iv) (3k – h)( 9k2 + 3kh + h2)                 (v) (2a + 3b)(4a2 – 6ab + 9b2)  

(vi) (4g – 5)( 16g2 + 20g + 25) (vii) (2p – 3q)(4p2 + 6pq + q2)

(viii) (5r – 3s)(25r2 + 15rs + 9s2)          (ix) 2(m + 2n)(m2 – 2mn + n2)

(x) 3(3a – b)(9a2 + 3ab + b2)                (xi) 5(2k – h)( 4k2 + 2kh + h2)

(xii) 3(2x + 3y)(4x2 – 6xy + 9y2)           (xiii) 5(3c + 2d)(9c3 – 6cd + 4d2)

(xiv) 2a(bc + d)(b2c2 – bcd + d2)          (xv) 3w(xy – 2z)(x2y2 + 2xyz + z2)

 

EQUATIONS

Q.1.     Solve the following equations.

(i)                  5x = 20

(ii)                4a + 7 = 31

(iii)               5g –3 = 22

(iv)              6p + 4 = 2p + 24

(v)                7b – 12 = b + 30

 

Q.2.     Write the following as equations and solve them.

 

(i)                  The sum of a number and 9 is equal to twice the number.

(ii)                The product of a number and 2 is equal to 14.

(iii)               The product of a number and 3 is equal to the sum of the number and 12.

(iv)              The sum of two consecutive numbers is 13.

(v)                The sum of three consecutive numbers is 27.

 

Q.3.     The length of a rectangle is 4 cm longer than its breadth. The perimeter is 28 cm. Write an equation to show the perimeter of the rectangle in terms of its breadth and solve the equation to determine the length and breadth of the rectangle.

 

Q.4.     Betty is 20 years older than her daughter, Jan. The sum of their ages is 26. How old are Betty and Jan?

 

Q.5.     The angles of a triangle are 2xo, 3xo and 4xo. What are the angles?

 

Q.6.     Solve for x :

(i)  + 4 = 6                                  (ii)  + 6 = 12

            (iii)  +  =                             (iv)  +  = 1

 

Q.7.     Solve the following inequations.

(i)                  2(x + 3) > 8                             (ii)  +  > 24

(iii)                <                          (iv)  <

 

ANSWERS:

Q.1.     (i) x = 4,           (ii) a = 6           (iii) g = 5          (iv) p = 5          (v) b = 7

Q.2.     (i) x + 9 = 2x    x = 9    (ii) 2x = 14       x = 7    (iii) 3x = x + 12            x = 6

            (iv) x + (x + 1) = 13     x = 6    numbers are 6 & 7

(v) x + (x + 1) + (x + 2) = 27               x = 8    numbers are 8, 9 & 10.

Q.3.     P = 2B + 2(B + 4) = 28                       B = 5   L = 9

Q.4.     Betty = 23, Jan = 3

Q.5.     40o, 60o, 80o.

Q.6.     (i) x = 4            (ii) x = 10         (iii) x = 1/5       (iv) x = 7 ½

Q.7.     (i) x > 1            (ii) x > 16         (iii) x < 9          (iv) x < -2

 

RESTAURANT ARITHMETIC & ALGEBRA

 

1.         A customer ordered a mixed grill at $17.90, a beef burger at $9.50 and a kid’s chicken schnitzel at $6.00 and handed Lorraine a $50 note. How much change would Lorraine give the customer?

2.         Wendy brought her family to the restaurant on her night off. She and her husband ordered roasts at $11-90 each while her two sons ordered steaks at $15.90 each. They also ordered 4 desserts at $4.50 each and two coffees at $2.50 each. How much change would Wendy get from $100?

3.         Jessica brought the customer a 300gram steak. If the density of the steak was 2.3g/ml, what was the volume of the steak? (answer to nearest ml)

4.         Hillary and Nalia were cleaning tables. Nalia cleaned 97 tables in 2 ½ hours while Hillary cleaned 129 tables in 3 ½ hours. Which of the two was working at a faster rate?

5.         The group ordered a number of plates of tortellini at $11.90 each and a number of garlic breads at $3.70 each. “That comes to exactly $110” said Jean. How many plates of tortellini and how many garlic breads did the group order?

6.         Barbara was explaining to the customer that a cup of coffee was 25 cents more than a pot of tea. She also said that for the same price the customer could purchase 10 pots of tea or 9 cups of coffee. What is the price of the tea and coffee?

7.         Nalia told the customer that three serves of fish and chips plus two desserts would cost $43.50 but two serves of fish and chips and three desserts would cost $36.50.

            What is the price of a serve of fish and chips and what is the price of a dessert?

8.         Hillary told the customer that a well-done steak would cook for twice the time as a rare steak. A steak cooked to “medium” takes 9 minutes which is also the average time for a well-done steak and a rare steak.

            What are the cooking times for rare steaks and well-done steaks?

9.         The menu board price for a kid’s cheese burger was higher than the price listed on the table menu. At the customer’s insistence, Mary charged the customer the $6.00 listed on the table menu. “Wow, this means that I pay 76.923% of the menu board price” said the customer. What was the price listed on the menu board?

10.       It was a busy night and Jo was “flat out” cleaning tables. She spent 30% of her time stationary at the tables and the rest of the time walking at an average speed of 4 km/h. If Jo worked from 5.30 p.m. to 9.30 p.m. how far did she walk during the night?

Answers:

1.         $16.60             2.  $21.40        3.  130 ml

4.         Nalia (38.8 tables/hr compared to 36.9 tables/hr for Hillary)

5.         8 tortellini, 4 garlic bread

6.         tea = $2.25, coffee = $2.50

7.         Fish & chips = $11.50, Desserts = $4.50

8.         Rare = 6 mins, well-done = 12 mins.

9.         $7.80

10.       11.2 km

 

  

   

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Suppose it can be factorised to (x - a)(x - b)

Then x2 + bx + c = (x - a)(x - b)