Stepwise Guide to Dividing Monomials and Polynomials
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 Explained
1. What is the division of algebraic expressions in mathematics?
The division of algebraic expressions is the process of breaking down a polynomial (the dividend) by another polynomial (the divisor) to find a result known as the quotient. It is the inverse operation of multiplication. For instance, if a × b = c, then c ÷ b = a. This process is fundamental for simplifying complex expressions and solving algebraic equations.
2. What are the main methods for dividing algebraic expressions as per the CBSE syllabus?
There are three primary methods for dividing algebraic expressions, depending on the complexity of the terms involved:
- Division of a Monomial by another Monomial: This involves dividing the numerical coefficients and subtracting the exponents of the variables.
- Division of a Polynomial by a Monomial: Each term of the polynomial is individually divided by the monomial.
- Division of a Polynomial by another Polynomial: This is typically done using the long division method, which is similar to the long division of numbers. Factorisation can also be used if the divisor is a factor of the dividend.
3. How do you divide a polynomial by a monomial with an example?
To divide a polynomial by a monomial, you must divide each term of the polynomial by the monomial separately. For example, to divide (9x³ + 6x²) by 3x, you would perform two separate divisions: (9x³ ÷ 3x) + (6x² ÷ 3x). This simplifies to 3x² + 2x. This method essentially distributes the division across all terms of the dividend.
4. What is the long division method for algebraic expressions and when is it used?
The long division method is a systematic procedure used to divide a polynomial by another polynomial, especially when the divisor is a binomial or has more terms. The steps are:
- Arrange the terms of both the dividend and the divisor in descending order of their powers.
- Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
- Multiply the entire divisor by this first term of the quotient and subtract the result from the dividend.
- Bring down the next term from the dividend to form a new dividend, and repeat the process until the degree of the remainder is less than the degree of the divisor.
5. What does the remainder signify in the division of algebraic expressions?
The remainder is the polynomial that is 'left over' after the division process is complete. If the remainder is zero, it signifies that the divisor is a perfect factor of the dividend. If the remainder is non-zero, it means the dividend is not perfectly divisible by the divisor. This concept is captured by the division algorithm: Dividend = (Divisor × Quotient) + Remainder.
6. Why is factorisation considered a crucial first step before attempting to divide polynomials?
Factorisation simplifies the division process significantly. By expressing the dividend and divisor as products of their factors, you can often cancel out common factors, which is much faster and less error-prone than the long division method. For example, dividing (x² - 9) by (x + 3) becomes much simpler by factoring the dividend into (x - 3)(x + 3). The (x + 3) terms cancel out, leaving the simple answer (x - 3). It reveals the underlying structure of the expressions before you begin the division.
7. What happens if you try to divide a polynomial by another polynomial of a higher degree?
When you divide a polynomial by another polynomial of a higher degree, the quotient is zero, and the remainder is the original dividend itself. For example, if you divide (x + 2) by (x² + 4x + 4), the process stops immediately because the degree of the divisor (2) is greater than the degree of the dividend (1). The result is not a polynomial but a rational expression, which is a fraction of two polynomials.
8. Is there a difference between finding the value of (x² - 4) ÷ (x - 2) and solving the equation (x² - 4) ÷ (x - 2) = 5?
Yes, there is a fundamental difference. Finding the value of (x² - 4) ÷ (x - 2) is about simplifying an expression, which results in another expression, (x + 2). Its value depends on the value of x. In contrast, solving the equation (x² - 4) ÷ (x - 2) = 5 means finding a specific numerical value for x that makes the statement true. In this case, you would simplify the expression to x + 2, set it equal to 5, and solve to find that x = 3. The first is simplification, while the second is finding a solution.
9. How is the division of algebraic expressions applied in real-world scenarios?
The division of algebraic expressions is a key tool for modelling and solving problems in various fields. For example:
- In business, it can be used to find the average cost per unit by dividing the total cost polynomial by the number of units polynomial.
- In physics, it helps in deriving formulas. For instance, if the formula for distance is a polynomial function of time, dividing it by time gives the expression for average speed.
- In engineering, it's used to simplify complex design equations and analyse signal processing models.
