How to Expand Algebraic Expressions with Examples
Expand Algebra
Can't wait to learn how to expand algebra or manipulate with algebraic expressions and/or simplify expressions. But what does this really mean? That said, there are certain rules to change the order of operations or expand an algebraic expression. For the purpose, we would combine more than one number or variable by conducting the assigned algebraic operation(s). We perform the action by using the distributive property in order to remove any brackets or parentheses and by combining like terms.
Introduction to Order of Operations
Same as the elation of driving a great car will not happen if you did not know how to begin it. True mathematics cannot occur without following some basic, yet significant rules. Those rules are actually the order of operations. An acronym PEMDAS stands for:
P - Parentheses (Brackets)
E - Exponents
M & D - Multiplication and Division as they happen- left to right
A & S - Addition and Subtraction as they happen - left to right
This is the order you would require to follow when assessing any algebraic expression. Needless to say, you might not have all these operations at the same time in the same expression.
How to Expand Algebraic Expression Involving Multiplication
In order to expand expressions that include multiplication, follow the rules of the distributive property which implies that any number can be multiplied by any number. Thus, numbers can be multiplied by another number, by itself or by a variable.
When you expand terms by distribution, you would require combining like terms for the purpose of simplification. Like terms are numbers from the similar group (4, 0, 5, or 89) or them sharing the same exponent and variable (5x2 and 7x2 are like terms). Let's take a look at expansion examples for clear understanding.
Example
Let's begin an easy expansion applying the distributive property:
7 (y+ 5) Using the distributive property
5 * y + 5 * 3 Multiply
5y + 15
Applying the postulate of PEMDAS, we begin with expanding the brackets or the parentheses. Seeing that the numbers in the brackets are not like terms (the y is a variable and the 5 is a number), we are unable to combine them by addition, and there were not even any exponents. Thus, we then applied the distributive property in order to multiply everything inside the bracket by everything on the outside. Hence, we multiplied both the y and the 5 within the parentheses by 7.
Sign Rules For Expanding
Don't forget the following sign rules for multiplication and division:
When two signs are similar - the outcome is positive
When two signs vary - the outcome is negative
Expanding Brackets
Expanding brackets implies to multiply each term in the bracket by the equation outside the bracket. For instance, in the expression 3 (m + 5), multiply both and 5 by 3, thus: 3 (m + 5) = 3 × m + 3 × 5 = 3 m + 15.
Expanding Two Sets of Brackets
For expanding two sets of brackets or parentheses, you would require to multiply each term in the 1st bracket by each term in the 2nd. Then, you will have to combine like terms. Don't skip seeing the signs!
Solved Examples on Expand Form
Example: (2a + 5) (3a - 4)
Solution:
Using the application of the distributive property:
= (2a) (3a) + (2a) (- 4) + (5) (3a) + 5 (-4)
= 6a2 - 8a + 15a – 20
= 6a2 + 7a – 20
Since - 8a and 15a are similar terms; we can combine terms to get 7a. In the example above, we were able to combine two of the terms in order to simplify the final answer.
Example: (3x + 4y + z) (2x – 3y)
Solution:
= (3x) (2x) + (3x) (-3y) + (4y) (2x) + (4y) (-3y) + (z) (2x) + (z) (-3y)
= 6x2 – 9xy + 8xy – 12y2 + 2xz – 3yz
= 6x2 – xy – 12y2 + 2xz – 3yz
Here, we will combine some terms in order to simplify the final answer. Note that the order of terms in the final answer does not have an impact on the accuracy of the solution.
Did You Know
While distributing a negative number, that negative sign will change the signs of each number that it is distributed to.
Usually, if an equation contains more than one variable, a polynomial is written in alphabetical order.
Special names are incorporated for some polynomials. A polynomial containing two terms is called a binomial.
FAQs on Expanding in Maths: Step-by-Step Guide
1. What does 'expanding' mean in the context of mathematics?
In mathematics, expanding refers to the process of multiplying out the parts of an algebraic expression that are contained within brackets or parentheses. The goal is to remove the brackets and write the expression as a sum or difference of individual terms. This is typically achieved using the distributive property.
2. What is the general method for expanding an algebraic expression like (a+b)(c+d)?
The general method for expanding expressions involves the distributive property. You must multiply each term in the first set of parentheses by every term in the second set. For an expression like (a+b)(c+d), the process is as follows:
- First, multiply 'a' by 'c' and then by 'd' to get ac + ad.
- Next, multiply 'b' by 'c' and then by 'd' to get bc + bd.
- Finally, combine all the results: ac + ad + bc + bd.
This ensures every term is correctly multiplied out.
3. How do standard algebraic identities, like (a+b)², help in expanding expressions more quickly?
Standard algebraic identities are essentially shortcuts for expansion. Instead of performing the full multiplication process, you can use these pre-proven formulas to get the answer directly. For example, to expand (x+5)², you could write it as (x+5)(x+5) and multiply. However, using the identity (a+b)² = a² + 2ab + b², you can instantly get the expanded form x² + 2(x)(5) + 5², which simplifies to x² + 10x + 25. This saves time and reduces the chance of calculation errors.
4. What is the key difference between expanding an algebraic expression and factoring it?
Expanding and factoring are opposite processes. Expanding involves removing parentheses by multiplication to write an expression as a single sum of terms. For example, expanding 2(x + 3) gives 2x + 6. On the other hand, factoring involves finding common factors to place an expression into parentheses. For example, factoring 2x + 6 gives 2(x + 3). In short, expanding removes brackets, while factoring introduces them.
5. Can you provide an example of expanding a binomial by a trinomial?
Certainly. To expand a binomial like (x + 2) by a trinomial like (x² + 4x - 5), you apply the distributive property. Each term of the binomial multiplies every term of the trinomial:
- Multiply x by the trinomial: x(x² + 4x - 5) = x³ + 4x² - 5x
- Multiply 2 by the trinomial: 2(x² + 4x - 5) = 2x² + 8x - 10
- Combine like terms: x³ + (4x² + 2x²) + (-5x + 8x) - 10
The final expanded expression is x³ + 6x² + 3x - 10.
6. Why is expanding expressions an important step before simplifying or solving complex equations?
Expanding is a crucial step because it converts an expression from a product form (with brackets) into a sum or difference of terms. This transformation is important for several reasons:
- Combining Like Terms: Once expanded, you can identify and combine like terms (e.g., all terms with x²), which is the primary method of simplification.
- Solving Equations: To solve an equation, you often need to isolate a variable. Expanding the expression allows you to rearrange terms and simplify the equation into a standard form (like a quadratic equation) that can then be solved.
- Identifying Polynomial Degree: Expanding an expression reveals its true form, making it easy to determine its degree and other properties.
