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Algebraic fractions

1. Equivalent algebraic fractions

2. Simplifying algebraic fractions

3. Manipulating algebraic fractions

4. Multiplying and dividing algebraic fractions

5. Adding algebraic fractions

6. Subtracting algebraic fractions

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Algebraic fractions

The rules that apply to numerical fractions also apply to algebraic fractions.

For example, if we multiply or divide the numerator and the denominator of a fraction by the same number or term we produce an equivalent fraction.

3x4x2

and are examples of algebraic fractions.2a

3a + 2

For example,

3x4x2

=34x

=68x

=3y4xy

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

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Equivalent algebraic fractions

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Simplifying algebraic fractions

We simplify or cancel algebraic fractions in the same way as numerical fractions, by dividing the numerator and the denominator by common factors. For example,

Simplify 6ab3ab2

6ab3ab2

=6 × a × b

3 × a × b × b

2

=2b

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Simplifying algebraic fractions

Sometimes we need to factorize the numerator and the denominator before we can simplify an algebraic fraction. For example,

Simplify 2a + a2

8 + 4a

=a4

2a + a2

8 + 4a=

a (2 + a)4(2 + a)

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Simplifying algebraic fractions

Simplify b2 – 363b – 18

b2 – 36 is the difference

between two squares.

b2 – 363b – 18

=(b + 6)(b – 6)

3(b – 6)

b + 63

=

If required, we can write this as

63

=b3

+b3

+ 2

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Manipulating algebraic fractions

Remember, a fraction written in the form

a + bc

can be written asbc

ac

+

However, a fraction written in the form

ca + b

cannot be written ascb

ca

+

For example,

1 + 23

=23

13

+ but3

1 + 2=

32

31

+

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Multiplying and dividing algebraic fractions

We can multiply and divide algebraic fractions using the same rules that we use for numerical fractions.

In general, ab

× =cd

acbd

ab

÷ =cd

ab

× =dc

adbc

and,

For example,3p4

× =2

(1 – p)6p

4(1 – p)=

3

2

3p2(1 – p)

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23y – 6

÷ =4

y – 2

This is the reciprocal

of4

y – 2

23y – 6

×4

y – 2

23(y – 2)

×=4

y – 2

16

=

Multiplying and dividing algebraic fractions

2

What is2

3y – 6 ÷

4y – 2

?

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Adding algebraic fractions

We can add algebraic fractions using the same method that we use for numerical fractions. For example,

What is1a

+2b

?

We need to write the fractions over a common denominator before we can add them.

1a

+2b

=b + 2a

abbab

+2aab

=

In general,

+ =ab

cd

ad + bcbd

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Adding algebraic fractions

What is3x

+y2

?

We need to write the fractions over a common denominator before we can add them.

3x

+y2

=

=6 + xy

2x

+62x

xy2x

=

+3 × 2x × 2

y × x2 × x

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Subtracting algebraic fractions

We can also subtract algebraic fractions using the same method as we use for numerical fractions. For example,

We need to write the fractions over a common denominator before we can subtract them.

In general,

What is – ?p3

q2

– =p3

q2

– =2p6

3q6

2p – 3q6

– =ab

cd

ad – bcbd

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Subtracting algebraic fractions

What is – ?

–(2 + p) × 2q

4 × 2q3 × 42q × 4

2 + p4

32q

=–2 + p

432q

= –2q(2 + p)

8q128q

=2q(2 + p) – 12

8q4

6

=q(2 + p) – 6

4q

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Addition pyramid – algebraic fractions