# Lesson Notes By Weeks and Term - Senior Secondary School 2

ALGEBRAIC FRACTIONS

SUBJECT: MATHEMATICS

CLASS:  SS 2

DATE:

TERM: 2nd TERM

REFERENCE BOOKS

• New General Mathematics SSS2 by M.F. Macraeetal.
• Essential Mathematics SSS2 by A.J.S. Oluwasanmi.

WEEK FOUR    DATE: __________

TOPIC: ALGEBRAIC FRACTIONS

CONTENT

-Simplification of Algebraic Fractions.

- Operation of Algebraic Fractions.

SIMPLIFICATION OF ALGEBRAIC FRACTIONS.

To simplify an algebraic fraction:

1. Factorise the numerator and the denominator of the fraction, where possible.
2. Divide the numerator and the denominator by the common factors. This process is sometimes known as cancelling a fraction. When a fraction cannot be reduced any further, we say the fraction is in its lowest or simplestform.

When simplifying a fraction, remember the following facts:

1. x2 – y2 = (x + y)(x + y) – difference of two squares.
2. (x + y)2 = x2 + 2xy + y2

(x – y)2 = x2 – 2xy + y2    (Perfect Squares)

1. x- y = -xy
2. -mn = -mn
3. -x-y = xy
4. xy = my x = m
5. To factorise x2 – 5x + 6, we have:

X2 – 5x + 6 = x2 -2x – 3x + 6

= x(x - 2) – 3(x - 2)

= (x – 2)(x – 3)

Example 1

Simplify the following fractions:

(a) 3x2+9x2y23x2y        (b) x2-y2+3x+3yx-y+3

(c) x2-9x2+x-6        (d) 5xy-10x+y-2 8-2y2

Solution

(a)3x2+9x2y23x2y = 3x2(1+3y)3x2 × y

Cancel the common factors

3x2(1+3y)3x2 × y = 1+3y2y

(b) x2-y2+3x+3yx-y+3 = x+yx+y+ 3(x+1)x-y+3

x+y(x-y+3)x-y+3

= x + y

(c) x2-9x2+x-6 = x+3(x-3)(x+3) (x-2) = x-3x-2

(d) 5xy-10x+y-2 8-2y2 = 5xy-2+ (y-2) 2(4-y2)

= y-2(5x+1) 22-y(2+y)

= y-2(5x+1) 22-y(2+y)

= -(5x+1) 2(2+y)

Notice that in the above y – 2 = - (2 – y)

In general: x – y = -(y – x)

e.g. 10 – 4 = -(4 – 10)

i.e. 6 = -4 + 10

6 = 6

Example 2

Simplify the following fractions:

(a) x2+9x+8 x2+6x+5        (b) 6x2+30x+36 2x2+12x+16

(c) 5x2-5x-100 4x2-8x-96        (d) (6x-18y)227y2-3x2

Solution

(a) x2+9x+8 x2+6x+5 = (x+8) (x+1) (x+5) (x+1) = x+ 8 x+5

(b) 6x2+30x+36 2x2+12x+16 = 6(x2+5x+6) 2(x2+6x+8)

Now factorise the quadratic expressions inside the brackets:

= 3x+3(x+2) x+4(x+2) = 3x+3(x+4)

(c) 5x2-5x-100 4x2-8x-96 = 5(x2-x-20)4(x2- 2x-24)

Now factorise the quadratic expressions inside the brackets:

= 5x+4(x-5)4x-6(x+4) = 5(x-5) 4(x-6)

(d) (6x-18y)227y2-3x2 = 6x-18y(6x-18y) 3(9y2-x2)

= 36x-3y(x-3y) 33y-x(3y+x)

But x – 3y = -(3y - x)

= 123y-x(x-3y) 33y-x(3y+x)

= 12(x-3y) 3y+x

Algebraic Fractions: Simplification, Operation and Undefined Fractions.

EVALUATION

1. 3x-93x 2. 8a4-24x312ax3-4a5        3. x2+3x-10x2+6x+5
2. 5x3y+15x2y5x2y2     5. xy2z-3x2y3zxy3 6. a3b3c4+abc2a3b2c2

OPERATION OF ALGEBRAIC FRACTIONS.

Multiplication and Division of Fractions

Factorise fully first, then divide the numerator and denominator by any factors that they have in common.

Example 1

Simplify a2+2a-3a2-16 × a+4a2+8a+15

Given expression

= a+3(a-1)a-4(a+4) × a+4a+5(a+3)

= a+1a-4(a+5)

The answer should be left in the form given.

Do not multiply out the brackets.

Example 2

Simplify m2-a2m2+bm+am+ ab ÷ m2-2am+a2cm+bc

To divide by a fraction, multiply by its reciprocal.

Given expression

=m2-a2m2+bm+am+ ab × cm+bcm2-2am+a2

=m-a(m+a)m+b(m+a) × c(m+b)m-a(m-a)

= cm -a

Example 3

Simplify

=a2+ ab a3-2ab+b3 ÷ a+3ba+2b × ab-a a2+3ab+2b2

Given expression

= a2+ ab a3-2ab+b3 × a+2ba+3b × ab-a a2+3ab+2b2

= a(a+b)a-b(a-b) × a+2ba+3b × a(b-a)a+b(a+2b)

= a2a-ba+3b

Notice that (a - b) divides into (b - a) to give -1.

This is because -1 x (a - b) = (b - a).

EVALUATION

1. 18ab15bc × 20cd24de     2. 12dn315cd3 ÷ 9c3n10c2d2        3. mn3m+3n
2. uv3u-6v × 4u-8vu2v 5. a-ba+ab ÷ 2a-2bab

Example 1

Simplify 6a- 32b

The denominators are a and 2b. The LCM of a and 2b is 2ab. Express each fraction with denominator of 2ab.

6a- 32b= 6 ×2ba ×2b- 3 ×a2b ×a

= 12b2ab- 3a2ab

= 12b-3a2ab

Example 2

Simplify 2 + 6a2+2b23ab- 4a-b2b

The denominators are 3ab and 2b. the LCM of 3ab and 2b is 6ab. Express each fraction in the expression with a denominator of 6ab.

2 + 6a2+2b23ab- 4a-b2b

= 2 ×6ab6ab+ 2(6a2+2b2)6ab- 3a(4a-b)6ab

12ab+12a2+4b2-12a2+3ab6ab

= 15ab+4b26ab

= b(15a+4b)6ab

= 15a+4b6a

Example 3

Simplify x+4x2-3x- x-19-x2

x+4x2-3- x-19-x2

= x+4x(x-3)- x-13-x(3+x)

= x+4x(x-3)- x-1x-3(3+x)

= x2+7x+12+x2-xxx-3(x+3)

= 2x2+6x+12xx-3(x+3)

= 2(x2+3x+6)xx-3(x+3)

Notice that the sign in front of the fraction is changed since (3 – x) = -(x – 3). This give an LCM of x(x – 3)(x + 3).

Example 3

Simplify 1a-3m- 2a+3m

1a-2m- 2a+3m= a+3m-2(a-2m)a-2m(a+3m)

= a+3m-2a+4ma-2m(a+3m)

= 7m-aa-2m(a+3m)

EVALUATION

Simplify the following.

1. 4x- 6x+2 2. 45d+ 73e        3. a+2a- 13ab
2. u2-v2uv+ vu- 3uv-u2v2

GENERAL EVALUATION/ REVISION QUESTIONS

Simplify the following.

1. x-327- 3x2                         2.mn+my2 mn-m
2. d+12d-8- d+212-3d 4. 2a+1+ 3a+2
3. 2a-2b+2c8bc × 10abc5a-5b+c

WEEKEND ASSIGNMENT

Objectives

1. Simplify xyz2axyz A. za B. xyz    C. xyza   D. yz
2. Simplify    ac-acdac2   A.  a-dc   B. 1-dc   C. a-ca   D. d-1a
3. Simplify   x2-1x-1     A.  1x+1     B. 1x-1     C.   x+1x-1D.  x+1
4. Simplify    2e+2- 1e+3 A.   e-8e+2(e+3) B.  e-6e+2(e+3) C.  e+8e+2(e+3)  D.  3e+4e+2(e+3)
5. Simplify 7pq2r21pq3r   A. 1q B.  prq   C.  q3p D.  13q

Theory

Simplify the following.

1.(a) 7pq2r21pq3r                            (b)  p-qq2-y2            (c) 1-p2p2-1

1. (a)    n2-9n2-n × n2-3n+2n2+n-6 (b) m2-n2m2-2mn+n2 ÷ m2+mnn2-mn (c) a-ab-6ba+ab-6b × a2-ab-ab2a2-2ab-3b2