Introduction on Multiplication of Algebraic Expression
Algebraic expressions explain a set of operations that should be done following a specific set of orders. Such expressions consist of an amalgamation of integers, variables, exponents, and constants. When these expressions undergo the mathematical operation of multiplication, then the process is called the multiplication of algebraic expression. Two different expressions that give the same answer are called equivalent expressions. Some other properties like distributive and commutative properties of addition will come in handy while multiplying polynomials. We will discuss the multiplication of algebraic expressions later, but first, we need to understand some terms used in algebra.
What are the Algebraic expressions in maths?
In mathematics, the equations or the expressions that consist of at least one variable and a constant can be considered as algebraic expressions. In these algebraic expressions, the various variables and constants are connected by different basic mathematical operations namely addition and subtraction.
For example, 2 x + 9 y, 3 y2, and many more equations or expressions like this are an example of algebraic expressions.
In the example of 2 x + 9 y, the mathematical numbers of 2 and 9 are considered as the constant, while the alphabetic letters of x and y are said to be the variables.
Algebraic Terms
The parts or terms of an algebraic expression consist of the following.
Integers: An integer is any positive, negative, or zero number, but it has to be whole numbers (not a fraction or decimal).
Variables: When alphabets or symbols are used in a mathematical problem to represent a specific value, then they are called variables.
Exponents: The exponent in mathematical expressions is the number that represents the number of times the quantity has been multiplied by itself. It is also called the power or indices of quantity. For example: In m3
The number 3 is an exponent, and the term represents that ‘m’ is raised to the power of 3.
m3
is equal to (m*m*m).
Constant: The terms in an algebraic expression that comprises only numbers (no variables) are called constants, for example, in the expression 4x - 5y + 7, the constant term is 7.
Polynomial Expression
The algebraic expression involving one or more terms, which comprise of variables, coefficients, exponents, and constants and combined by using mathematical operations like addition, subtraction, multiplication, and division, is called a polynomial.
An expression is only considered to be polynomial in the absence of the following elements- fraction power of the variable, negative exponents of a variable, square-roots of variables, and variables in the denominator.
The mentionable types of polynomial expressions are Monomial, Binomial, and Trinomial, and their names are such because the expressions involve one, two, and three terms, respectively.
Monomial expression: The algebraic expression involves only one term formed by the combination of integer and variable, for example: 3x, 4a, 5b, 3abx, etc. where x, a, b are the variables and 3, 4, 5 are called the coefficients.
Binomial expression: The polynomial expression involves two terms, for example, 2x - 1, xy - 5z2, etc.
Trinomial expression: The algebraic expression that involves three terms, for example, 5x + 3y - 2, 7y2 + 9y + 11, etc.
Multiplying Algebraic Expressions:
While doing the multiplication of algebraic expressions, one should know about the proper operations of addition, subtraction, multiplication, and division of numeric values and variables.
Some rules that must be remembered while multiplying algebraic expression are:
The product of two factors with the same signs will be positive, and the outcome of multiplying two terms with two, unlike signs, will be negative.
If x is variable and a, b are positive integers then, (xa * xb) = x(m +n)
Multiplication of Monomial by Monomial :
The multiplication of two or more monomial expressions or expressions with one term means finding the product of all the expressions involved. While multiplication of monomials by monomial expressions the rule or equation that applies is mentioned below.
The product of monomials = (product of their coefficients)*(product of the variables).
Multiplication of Algebraic Expressions Examples:
a*a = a2
2a*2b = (2*2)*(a*b) = 4ab
6ab*3x = (6*3)*(ab*x) = 18abx
5xy * 4x2 * 2x3 = (5*4*2)*{x(1+2+3) * y} = 40yx6
3x * (-5a2)(2b3) = (3 * 2) * (-5) * (x * a2 * b3) = -15 x a2 b3
Multiplying Monomials and Polynomials:
The rule that applies to the multiplication of monomials and polynomials is the distributive law.
The law shows that each term of the polynomial should be individually multiplied by the monomial expression, x*(y +z) = (x*y) +(xz) = xy +xz and x*(y –z) = xy - xz
Multiplication of Monomials and Polynomial Examples:
a*(a +b) = a2 +ab
4xy(3xy) = 12(xy)²
(-ab)*(a –b +c) = -a2b +ab2 –abc
3xyz(x +2y -3) = 3yzx2 +6xzy2 -3xyz
5xy2*(3x+7y) = (5*3)*x(1 +1)*y² +(5*7)*x*y(2 +1) = 15x2 y2 + 35xy3
Multiplication of Two Binomials
An expression is said to be a binomial when they are made up of two individual terms or monomials.
In this article, students get knowledge on the various aspects of the multiplication of algebraic expressions. The various terms used in the concept of the multiplication of various algebraic expressions, different types of algebraic equations, and their multiplication with the same kind of mathematical equation or the other categories of algebraic expressions, like the multiplication of monomial with monomial, binomial with binomial, polynomials with the monomials, or various other types of multiplication, are explained here.
FAQs on Multiplication of Algebraic Expression
1. What is an algebraic expression and what are its main types?
An algebraic expression in mathematics is a combination of constants, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. For example, in the expression 5x + 3, 'x' is the variable, '5' is the coefficient, and '3' is the constant. Based on the number of terms, they are typically classified as:
- Monomial: An expression with a single term (e.g., 7xy).
- Binomial: An expression with two terms (e.g., 2a - 3b).
- Trinomial: An expression with three terms (e.g., x² + 2x + 1).
- Polynomial: An expression with one or more terms with non-negative integer exponents on the variables.
2. What is the fundamental rule for multiplying algebraic expressions?
The fundamental rule for multiplying algebraic expressions is the distributive property. This means you must multiply each term in the first expression by each term in the second expression. For example, to multiply a monomial by a polynomial, you distribute the monomial to every term inside the polynomial. For multiplying two polynomials, every term of the first polynomial is multiplied by every term of the second.
3. Can you show an example of multiplying a binomial by another binomial?
Certainly. Let's multiply the binomial (x + 2) by (x + 5). We use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last).
- First: Multiply the first terms: x * x = x².
- Outer: Multiply the outer terms: x * 5 = 5x.
- Inner: Multiply the inner terms: 2 * x = 2x.
- Last: Multiply the last terms: 2 * 5 = 10.
Now, combine the like terms (5x and 2x): x² + 5x + 2x + 10 = x² + 7x + 10. This is the final product.
4. How are the rules for signs applied when multiplying algebraic terms?
The rules for signs are crucial for getting the correct answer. They are the same as in arithmetic:
- Multiplying two terms with the same sign (both positive or both negative) results in a positive product. (e.g., (+a) * (+b) = +ab and (-a) * (-b) = +ab).
- Multiplying two terms with different signs (one positive and one negative) results in a negative product. (e.g., (+a) * (-b) = -ab).
Always determine the sign of the resulting term before multiplying the coefficients and variables.
5. What are some real-world applications of multiplying algebraic expressions?
Multiplication of algebraic expressions is widely used in various practical scenarios. For instance:
- Calculating Area: If a rectangular garden has a length of (x + 5) metres and a width of (x + 2) metres, its area is found by multiplying the two expressions: (x + 5)(x + 2) = x² + 7x + 10 square metres.
- Business and Finance: To calculate total revenue, you might multiply the price per item (which could be an expression like 100 - p) by the quantity sold (an expression like 500 + 10p).
- Physics: Formulas involving distance, speed, and time often require multiplying expressions, especially when quantities change over time.
6. What is the main difference between adding and multiplying algebraic expressions?
The main difference lies in how we treat the terms. When adding or subtracting, you can only combine 'like terms' (terms with the same variables raised to the same power). For example, 3x + 2x = 5x, but 3x + 2y cannot be simplified further. When multiplying, you can multiply any two terms, regardless of whether they are like or unlike. For example, (3x) * (2y) = 6xy. The exponents of the variables are added during multiplication (x * x = x²), whereas they remain unchanged during addition (x + x = 2x).
7. Why does the number of terms in the product sometimes seem less than expected before simplification?
Before simplification, the number of terms in the product of two polynomials is the product of the number of terms in each polynomial. For example, multiplying a binomial (2 terms) by a trinomial (3 terms) will initially result in 2 * 3 = 6 terms. The final expression often has fewer terms because the process of multiplication creates 'like terms' that can be combined. For example, in (x+1)(x+2) = x² + 2x + x + 2, the terms '2x' and 'x' are like terms that combine to give 3x, reducing the final expression to three terms (x² + 3x + 2).
8. What are the most common mistakes to avoid when multiplying algebraic expressions?
Students often make a few common errors. Being aware of them can greatly improve accuracy:
- Forgetting to Distribute: When multiplying a monomial by a polynomial, a common error is to multiply only the first term of the polynomial. For example, incorrectly calculating 3(x + 4) as 3x + 4 instead of the correct 3x + 12.
- Sign Errors: Mismanaging negative signs is a frequent mistake. Forgetting that multiplying two negatives gives a positive (e.g., (-x) * (-y) = xy) is a major pitfall.
- Exponent Errors: Confusing the rules for exponents. When multiplying variables, you add their powers (e.g., x² * x³ = x⁵), you do not multiply them.
9. How would you approach multiplying three algebraic expressions together?
To multiply three or more algebraic expressions, you should work in steps. First, multiply any two of the expressions using the distributive property and simplify the result. Then, take this new polynomial product and multiply it by the third (remaining) expression. For instance, to calculate (x + 1)(x + 2)(x + 3), you would first find the product of (x + 1)(x + 2), which is x² + 3x + 2. Then, you multiply this result by (x + 3): (x² + 3x + 2)(x + 3).
