H.C.F of algebraic expressions

From our elementary schools, we have been told algebra is the combination of letters, numbers and symbols in mathematical expressions. Examples of algebraic expressions are xy, 2a2b, 3αβ, 7α2β3 etc.

Example

Find the H.C.F of the following:

          1.     ab and bc

Solution

The index and table methods are the best to use when finding the H.C.F of algebraic expressions.

Index method

as we can see from above, b is common to the two expressions

H.C.F = b

Table method

Just like our normal H.C.F, we look for what can divide the two expressions

b ab bc
  a c

** Since a is not common to the two expressions, check b, b is common. We will now use the b to divide the expressions one by one.

** . The b at the denominator will cancel that of the numerator, leaving only a at the numerator.

** The a remaining will be written in the next row under ab.

** . b will cancel b, leaving c at the numerator, The c remaining will be written in the next row under bc.

** we look at the latest row, it has only a and c, nothing is common to this 2 expressions again, then we conclude that our H.C.F is b

 

          2.    x2y and x3y2

Solution

Index method

bases x and y are common to the two expressions, so we pick the one with the smallest power.

Between x2 and x3, we pick x2

Between y and y2, we pick y (the one with the smallest power)

H.C.F = x2 x y

= x2y

Table method

x x2y x3y2
x xy x2y2
y y xy2
  1 xy

** x is common to x2y and x3y2. So we use the x to divide each.

** When used to divide x2y, the x will cancel out one x from the x2, remaining only one x and y. the xy will be brought to the next row under x2y.

** when used to divide x3y2, it will cancel one x from x3, leaving x2 and y2. The x2y2 remaining will be brought to the next row under x3y2.

** we do the same for row 2, and so on till no letter is common.

** H.C.F is the product of the letters in the first column.

HCF = x X x X y

= x2y

 

          3.    8p2q3, 12pq2 and 20p2q3

Solution

Index method

Express the numbers as the product of their prime factors in index form first and use them to multiply the letters.

Note: select only the base that is common to the three expressions, and pick the one with the smallest power.

Tabular Method 

2 8p2q3 12pq2 20p2q3
2 4p2q3 6pq2 10p2q3
p 2p2q3 3pq2 5p2q3
q 2pq3 3q2 5pq3
q 2pq2 3q 5pq2
  2pq 3 5pq

HCF = 2 x 2 x p x q x q

= 4pq

 

                   Exercise F

Find the HCF of the following

1. 6pq and 4q

2. 18x and 15y

3. 81pt, 63p and 45pq

4. 3m, mn and 5m

5. 14a2 and 21ab2

6. 12x2y, 15x3y2 and 9x2y2

7. 4uv and 3ab

8. 24b2c3d, 18b3c5d4 and 30b5c5d5

9. 27jk and 18jp

10. 45g2h3k5, 99g3h3k3, 27g4h2k

 

 

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