How to Divide Algebraic Expressions Using Formula and Step by Step Examples
Numbers are the basic units to accurately express a quantity in mathematical terms. Along with this even alphabets are used in cases of problem-solving, decision-making, and word problems. And, variables are symbols used in denoting numbers, alphabets and other arbitrary elements. Combining all, when a statement or a mathematical condition has numbers, alphabets, variables, or any other special character, then this is termed to be as an Algebraic expression. Addition, subtraction, multiplication, and division are the 4 operations that function as an algebraic expression.
From here on, let’s learn the important pointers about the division of algebraic expression only with examples.
A Brief Idea about Division of Algebraic Expression
Any algebraic expression will consist of numbers, operations, variables and a few special characters used in the field of mathematics. Since the sequence is formed using constants and variables, this formation is referred to as a mathematical expression.
Note that no algebraic expression is found to have equal signs or sides. Examples for algebraic expressions are 4x + 5y – 3, 2x – 5, 4x2 – 1xy + 6, 6x3 – 2x2 + 6x – 1 and so on.
Lastly, the division of algebraic expression is the inverse proportion of multiplication. In the classification of algebraic expressions, there are 3 majors namely Monomial, Binomial and Polynomial.
We are going to understand each concept in short as described in the following sections.
Defining what is a Monomial and a Polynomial
Polynomial is the term used to denote the condition of P = 0. Here, this formula is also called the Polynomial equation or Polynomial expression. A polynomial can range anywhere between a trinomial, binomial or even possess ‘n’ count of terms.
When a polynomial formula has only 1 term, then this is said to be the monomial. 15xy is an example of a monomial term.
Binomial, as the name says, is a polynomial equation that has only 2 terms. To state an example, 1x + 4. Take the case of 2a(a+b) 2. This is also considered as a binomial expression due to the presence of binomial factors ‘a’ and ‘b’.
The Process of Dividing a Monomial by Another Monomial
Consider this division of polynomial expression 12a3 ÷ 4a. Here, the terms 4a and 12a3 are the 2 monomials of the equation. One can quickly simplify the equation by cancelling common values. This is quite similar to that of natural division in real numbers.
12a3 ÷ 4a = (12 × a × a × a) / 4 × a
After the cancellation process, we will get 12a3 ÷ 4a = 3a2.
When a Monomial Divides a Polynomial
As we read before, a polynomial equation can have multiple options such as 2, 3 or ‘n’. For this condition, we are considering a trinomial. We will take a monomial “3a” and then divide it using the polynomial equation “(6a3 + 7a2 + 9a)”.
(6a3 + 7a2 + 9a) ÷ 3a
Take only the common factors from the equation and for our case, 3a is the common value. So the equation will note become:
(6a3 + 7a2 + 9a) = 3a (2a2 + 7/3a + 3)
Start dividing the complete set by the monomial 3a.
(6a3 + 7a2 + 9a) ÷ 3a = 3a (2a2 + 7/3a + 3) / 3a
Upon cancellation of the terms 3a and 3a from the denominator and the numerator, we will get:
(6a3 + 7a2 + 9a) ÷ 3a = (2a2 + 7/3a + 3)
Hence, we divided a polynomial expression using a monomial term. Finally, we are about to learn the division of algebraic expression between 2 different polynomial values as given below.
Division Operation Between 2 Polynomials
We will consider this case using simple numbers for a better understanding. Take 2 separate polynomial figures and divide them.
(6a2 + 12a) ÷ (a + 2)
Did you notice that the above 2 polynomials are in the monomial form? Now, pick the common factors, which is 6a, and the equation becomes:
(6a2 + 12a) = 6a (a + 2)
As you might have guessed, divide the complete set with the monomial (a + 2).
(6a2 + 12a) ÷ (a + 5) = 6a (a + 2) / (a + 2)
By cancelling the common terms (a +2), we get (6a2 + 12a) ÷ (a + 5) = 6a
Thus, we formed a proper equation, resulting in a monomial value.
Conclusion
An Algebraic Expression is defined as a mathematical sequence, formulated using operational symbols, numbers, constants and variables. The Division of Algebraic Expression refers only to diving between the various types of expressions. Monomials, Binomials, and trinomials are classified as Polynomials for an algebraic equation. Monomial has only 1 term and Binomial has 2 terms, while a polynomial can have ‘n’ count of values. The division can happen between 2 monomials, 2 binomials, 2 polynomials or even crossing 1 type from each other. Always, start the division process by identifying the common terms and cancelling them from the numerator and denominator.
FAQs on Division of Algebraic Expressions with Methods and Examples
1. What is division of algebraic expressions?
The division of algebraic expressions is the process of simplifying one algebraic expression by dividing it by another using factorization and cancellation of common factors. In simple terms, it works like fraction division but with variables.
- Write the division as a fraction.
- Factor both numerator and denominator completely.
- Cancel common factors.
- Simplify the remaining expression.
2. How do you divide algebraic expressions step by step?
To divide algebraic expressions, factor and cancel common factors to simplify the fraction. Follow these steps:
- Step 1: Rewrite the division as a fraction.
- Step 2: Factor the numerator and denominator completely.
- Step 3: Cancel common factors.
- Step 4: Multiply remaining terms if needed.
3. What is the formula for dividing algebraic fractions?
The formula for dividing algebraic fractions is (a/b) ÷ (c/d) = (a/b) × (d/c). This means you multiply by the reciprocal of the second fraction.
- Keep the first fraction the same.
- Change division to multiplication.
- Flip the second fraction.
- Simplify by canceling common factors.
4. How do you divide polynomials by monomials?
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial separately. Apply exponent laws while dividing.
- Divide coefficients.
- Subtract exponents of like bases: am/an = am−n.
5. How do you divide two polynomials?
Two polynomials are divided using long division or synthetic division depending on the divisor. Polynomial long division follows steps similar to numerical long division.
- Arrange terms in descending powers.
- Divide leading terms.
- Multiply and subtract.
- Repeat until remainder is obtained.
6. Can you give an example of dividing algebraic expressions?
An example of dividing algebraic expressions is simplifying \((x^2 − 9)/(x − 3). Factor the numerator first.
- \(x^2 − 9 = (x − 3)(x + 3)
- Cancel the common factor (x − 3).
- Result = x + 3 (for x ≠ 3).
7. What are the rules for dividing powers in algebra?
When dividing powers with the same base, subtract the exponents using the rule am/an = am−n. This is called the law of exponents.
- Same base is required.
- Subtract numerator exponent minus denominator exponent.
- If result exponent is zero, value is 1 (a ≠ 0).
8. What is the difference between dividing monomials and polynomials?
The difference is that monomial division involves single-term expressions, while polynomial division involves multi-term expressions. Monomial division is simpler and uses exponent rules directly.
- Monomial ÷ Monomial: Divide coefficients and subtract exponents.
- Polynomial ÷ Polynomial: Use long division or synthetic division.
9. What are common mistakes when dividing algebraic expressions?
A common mistake in dividing algebraic expressions is canceling terms instead of canceling factors. Only factors can be canceled.
- Wrong: Canceling terms in (x + 2)/x.
- Correct: Factor first before canceling.
- Forgetting to restrict values that make denominator zero.
- Not applying exponent subtraction rule correctly.
10. Why can’t we divide by zero in algebraic expressions?
You cannot divide by zero because division by zero is undefined in mathematics. Any algebraic expression with a denominator equal to zero has no value.
- If denominator = 0, expression is undefined.
- Always find restrictions by solving denominator = 0.
