Taking out common factors, and practising index laws

commonfactors

Index laws

Practice questions, and answers, here

Common factors

Watch this video for the basics: video on taking out common factors. Do these worksheets to get good at taking out common factors in algebra without fractions.

Common factors worksheet 1

Common factors worksheet 2

Common factors worksheet 3

When we’re doing series or induction problems, we often have to take out common factors involving fractions.

Example: 16n(n+1)(2n+1) + 12n(n+1)

Choose the common factor you take out so that all the fraction bit is now in the common factor, and in the big bracket there are only whole numbers.

So take 16n(n+1) as the common factor.

16n(n+1)(2n+1) + 12n(n+1) = 16n(n+1) [(2n+1)+3] … because 16n(n+1) divides into 12n(n+1) three times, same as 16 of anything divides into 12 of that same thing three times.

So

16n(n+1)(2n+1) + 12n(n+1) = 16n(n+1) [(2n+1)+3]
     = 16n(n+1) (2n+4)

Practice

Take out common factors and simplify these:

1. 16n(n+1)(2n+1) + n(n+1)

2. 16n(n+1)(2n+1) + 2n(n+1)

3. 14n2(n+1)2 + 16n(n+1)(2n+1) + n(n+1)

4. 24n2(n+1)2 + 16n(n+1)(2n+1)

5. 14n2(n+1)2 + 26n(n+1)(2n+1)

6. 14n2(n+1)2 + 36n(n+1)(2n+1) + n(n+1)

7. 16n(n+1)(2n+1) + n

8. 16n(n+1)(2n+1) − n

9. 16n(n+1)(2n+1) − 2n

10. 16n(n+1)(2n+1) + 12n(n+1) − n

11. 12(k+1)(k+2) − 12k(k+1)

12. 16(k+1)(k+2)(2k+3) − 16k(k+1)(2k+1)

13. 14(k+1)2(k+2)214k2(k+1)2

More practice:

Worksheet

Another worksheet