Algebraic Fractions
DILKI IYONA RATHNAYAKE
Learning Outcomes
End of this lesson you will be able to learn how to factorize the
algebraic expressions
As well as you can learn how we simplify the algebraic fractions.
Factor Identities
Examples
Factorise the following:
8x² - 32x = 8x(x – 4)
x² + 5x – 14 = (x + 7)(x – 2)
3x² + 17x + 10 = (3x + 2)(x + 5)
Simplifying Algebraic Fractions
To simplify algebraic fractions:
1) factorise the top and/or the bottom
2) cancel common factors Why can’t we
cancel this
3x + 6 = 3(x + 2) = x + 2 further?
3x² 3x² x²
4ab + 8a² = 4a(b + 2a) = b + 2a
12a 3 12a 3
Checkpoint
7x – 35 3x² + 15x
21 27x
x – 5 x + 5
3 9
6x²y + 18xy x² + 7x + 10
21xy 3x + 15
2(x + 3) x + 2
7 3
Simplifying Algebraic Fractions
To simplify algebraic fractions:
1) factorise the top and/or the bottom
2) cancel common factors
4x - 20 = 4(x - 5) = _4_
x² - 13x + 40 (x – 5)(x – 8) (x – 8)
x² - 16 = (x + 4)(x – 4) = (x – 4)
x² - x - 20 (x + 4)(x – 5) (x - 5)
Exercises
5x – 25 12x + 30 20x² - 40x
10x² 9x² 15x²
8ab – 16b 9cd² - 24d² 14efg² + 7ef²
24ab 15d 21efg
x² + 9x + 20 x² - 3x - 88 x² - 49
7x + 28 x² - 21x + 110 x² - 5x - 14
2x² + 7x + 3 5x² - 3x - 14 6x² - 13x - 5
x² - 9 x² - 11x + 18 10² - 19x - 15
Multiplying Algebraic Fractions
To multiply algebraic fractions:
1) cancel common factors
2) multiply the numerators, multiply the denominators
3
4a x 15b = a x 3b = 3ab
5 16 4 4 4
n + 4 x 3n – 9 = n + 4 x 3(n – 3) =3
n–3 5n + 20 n–3 5(n + 4) 5
Dividing Algebraic Fractions
To divide algebraic fractions:
1) flip the second fraction and change ÷ to x
2) cancel common factors
3) multiply the numerators, multiply the denominators
_11_ ÷ 77y = _11_ x 15x 5 = _1_ x _5_ = _5_
9x²y 15x 3x 9x²y 7 77y 3xy 7y 21xy²
x² - 4 ÷ x² - 5x + 6 = x² - 4 x 5x + 40
x² + 4x – 32 5x + 40 x² + 4x – 32 x² - 5x + 6
= (x + 2)(x – 2) x 5(x + 8) = 5(x + 2)
(x + 8)(x – 4) (x – 2)(x – 3) (x - 4)(x – 3)
Adding and Subtracting Fractions
2 + 1
3 4
1) Find a common denominator: 3 x 4 = 12
2) Find equivalent fractions with that denominator:
2 = 8 1 = 3
3 12 4 12
3) Add (or subtract!) the numerators and put the answer over the
same denominator:
8 + 3 = 11
12 12 12
4) Simplify if possible.
Adding and Subtracting Algebraic Fractions
To simplify algebraic fractions:
1) Find a common denominator
2) Find equivalent fractions with that denominator
3) Add (or subtract!) the numerators and put the answer over
the same denominator
4) Simplify if possible.
2x + x = 8x + 3x = 11x
3 4 12 12 12
4 - 5 = 8 - 5 = 3
x 2x 2x 2x 2x
Adding and Subtracting Algebraic Fractions
To simplify algebraic fractions:
1) Find a common denominator
2) Find equivalent fractions with that denominator
3) Add (or subtract!) the numerators and put the answer over
the same denominator
4) Simplify if possible.
x+1 + x–3 = 7(x + 1) + 4(x – 3)
4 7 28 28
= 7x + 7 + 4x + 12 = 11x - 5
28 28
Adding and Subtracting Algebraic Fractions
To simplify algebraic fractions:
1) Find a common denominator
2) Find equivalent fractions with that denominator
3) Add (or subtract!) the numerators and put the answer over
the same denominator
4) Simplify if possible.
_9_ - _8_ = 9(x + 4) - 8(x – 2)
x–2 x+4 (x – 2)(x + 4) (x + 4)(x – 2)
= 9x + 36 – 8x + 16 = x + 52
x² + 2x - 8 x² + 2x - 8
Algebraic Fractions – Adding and Subtracting
Exercises 1) 7 + 5
x x
2) 8x
3
+ 4x
3
3) 5x + 1 - 3x 4) 7x + 3x
2 2 3 4
5) 7x - 8x 6) 12 + 3
3 5 5x x
7) x + 3 + x - 2 8) 2x – 5 - 4x + 1
6 4 3 7
9) _7_ - _9_ 10) _3_ + _2_
x + 4 x - 2 x – 1 x + 1
Exercises
Thank you!
