Lesson 3, Topic 4
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# Complete Factorization

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To factorize an expression, divide each term of the expression by their common factor. To factorize completely, use the Highest Common Factor (HCF).

### Example 8

Factorize completely

(a) 2a + 4ab

(b) 5x + 10y

(c) 15mn2 â€“ 10m2n2

(d) $$\frac{2 \pi r}{T} \: + \: \frac{2 \pi r^2}{T^2} \: – \: \frac{2 \pi r^3}{T^3}$$

Solution

(a) 2a + 4ab

Step 1: find the HCF of 2a and 4ab

HCF = 2 x a = 2a

= 2a + 4ab

= 2a (1 + 2b) â†’ last

(b) 5x + 10y

Find the HCF of 5x and 10y

HCF = 5

5x + 10y = 5 (x + 2y) â†’ last

(c) 15mn3 â€“ 10m2n2

Find the HCF

HCF = 5 x m x n x n

= 5mn2

15mn3 â€“ 10m2n2 = 5mn2 (3n â€“ 2m)

(d) $$\frac{2 \pi r}{T} \: + \: \frac{2 \pi r^2}{T^2} \: – \: \frac{2 \pi r^3}{T^3}$$

H.C.F = $$\scriptsize 2 \: \times \: \pi \: \times \: r \: \times \: \normalsize \frac{1}{T}$$

H.C.F = $$\frac{2 \pi r}{T}$$

$$\frac{2 \pi r}{T} \: + \: \frac{2 \pi r^2}{T^2} \: – \: \frac{2 \pi r^3}{T^3}$$

= $$\frac{2 \pi r}{T} \left (\scriptsize 1 \: + \: \normalsize \frac{r}{T} \: -\: \frac{r^2}{T^2} \right)$$

Evaluation

Factorize completely each of the following

1. 4x2y  +  12xy2
2. 17m  –  34m2
3. 5k2  –  k
4. 35p  +  20p2
5. 9ab  + 3a
6. mn3  –  3gmn  –  g3m2n3
7. $$\frac{mv}{t} \: -\: \frac{mu}{t}$$

7. $$\frac{m}{t} \scriptsize (v \: -\: u)$$