Division of Algebraic Expressions

This topic gives an overview of;

- Division of Algebraic Expressions
- Division of a monomial by another monomial
- Division of a polynomial by a monomial
- Division of a polynomial by a Polynomial)

We have learnt how to add and subtract algebraic expressions. We also know how to multiply two expressions. We have not however, looked at division of one algebraic expression by another. This is what we wish to do in this section.

We recall that division is the inverse operation of multiplication. Thus, 7 × 8 = 56 gives 56 ÷ 8 = 7 or 56 ÷ 7 = 8.

We may similarly follow the division of algebraic expressions. For example,

2x * 3x^{2} = 6x^{3}

Therefore, 6x^{3} ÷ 2x= 3x^{2 } and also,6x^{3} ÷ 3x^{2} = 2x.

5x (x + 4) = 5x^{2} + 20x

Therefore,(5x^{2} + 20x) ÷ 5x = x + 4 and also(5x^{2} + 20x) ÷ (x + 4) = 5x.

We shall now look closely at how the division of one expression by another can be carried out. To begin with we shall consider the division of a monomial by another monomial.

Consider 6x^{3 }÷ 2x

We may write 2x and 6x^{3} in irreducible factor forms,

2x = 2 * *x*

6x^{3} = 2 * 3 × *x* **x* * *x*

Now we group factors of 6x3 to separate 2x,

6x^{3} = 2 **x* * (3 * *x* * *x*) = (2*x*) * (3*x*^{2}) Therefore, 6x^{3} ÷ 2*x* = 3*x ^{2}*

A shorter way to depict cancellation of common factors is as we do in division of numbers:

Let us consider the division of the trinomial 4y^{3} + 5y^{2} + 6y by the monomial 2y.

4y^{3} + 5y^{2} + 6y = (2 * 2 * y * y * y) + (5 * y * y) + (2 * 3 * y)

Here, we expressed each term of the polynomial in factor form) we find that 2 × y is common in each term. Therefore, separating 2*×y* from each term. We get

Therefore, (4y^{3} + 5y^{2} + 6y) ÷ 2y

Alternatively, we could divide each term of the trinomial by the monomial using the cancellation method.

- Consider (7x
^{2}+ 14x) ÷ (x + 2)

We shall factorise (7x^{2} + 14x) first to check and match factors with the denominator:

- We know that in the case of numbers, division is the inverse of multiplication. This idea is applicable also to the division of algebraic expressions.
- In the case of division of a polynomial by a monomial, we may carry out the division either by dividing each term of the polynomial by the monomial or by the common factor method.
- In the case of division of a polynomial by a polynomial, we cannot proceed by dividing each term in the dividend polynomial by the divisor polynomial. Instead, we factorise both the polynomials and cancel their common factors.
- In the case of divisions of algebraic expressions that we studied in this chapter, we have
- Dividend = Divisor × Quotient.
- In general, however, the relation is
- Dividend = Divisor × Quotient + Remainder

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