** 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, 2a^{2}b, 3αβ, 7α^{2}β^{3} 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. x^{2}y and x^{3}y^{2}

__Solution__

Index method

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

Between x^{2 }and x^{3}, we pick x^{2}

Between y and y^{2}, we pick y (the one with the smallest power)

H.C.F = x^{2} x y

= x^{2}y

Table method

x | x^{2}y |
x^{3}y^{2} |

x | xy | x^{2}y^{2} |

y | y | xy^{2} |

1 | xy |

** x is common to x^{2}y and x^{3}y^{2}. So we use the x to divide each.

** When used to divide x^{2}y, the x will cancel out one x from the x^{2}, remaining only one x and y. the xy will be brought to the next row under x^{2}y.

** when used to divide x^{3}y^{2}, it will cancel one x from x^{3}, leaving x^{2} and y^{2}. The x^{2}y^{2} remaining will be brought to the next row under x^{3}y^{2}.

** 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**

= x^{2}y

3. 8p^{2}q^{3}, 12pq^{2 }and 20p^{2}q^{3}

__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 | 8p^{2}q^{3} |
12pq^{2} |
20p^{2}q^{3} |

2 | 4p^{2}q^{3} |
6pq^{2} |
10p^{2}q^{3} |

p | 2p^{2}q^{3} |
3pq^{2} |
5p^{2}q^{3} |

q | 2pq^{3} |
3q^{2} |
5pq^{3} |

q | 2pq^{2} |
3q | 5pq^{2} |

2pq | 3 | 5pq |

HCF = 2 x 2 x p x q x q

= 4pq^{2 }

** 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. 14a^{2 }and 21ab^{2}

6. 12x^{2}y, 15x^{3}y^{2} and 9x^{2}y^{2}

7. 4uv and 3ab

8. 24b^{2}c^{3}d, 18b^{3}c^{5}d^{4} and 30b^{5}c^{5}d^{5}

9. 27jk and 18jp

10. 45g^{2}h^{3}k^{5}, 99g^{3}h^{3}k^{3}, 27g^{4}h^{2}k^{8 }

**PREVIOUS TOPIC: H.C.F OF NUMBERS **

**NEXT: Multiples, Common multiples, Least/Lowest Common multiples**