How to Expand Algebraic Expressions with Formula and 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 Algebraic Expressions Step by Step Guide
1. What does expanding mean in algebra?
Expanding in algebra means rewriting an expression by removing brackets using multiplication. In expanding expressions, each term inside the bracket is multiplied by the term outside the bracket.
For example:
3(x + 4)
Multiply 3 by both terms:
- 3 × x = 3x
- 3 × 4 = 12
2. How do you expand a single bracket?
To expand a single bracket, multiply the term outside the bracket by every term inside the bracket. This uses the distributive property.
Steps:
- Multiply the outside term by the first term inside.
- Multiply the outside term by the second term inside.
- Simplify if possible.
5(2x − 3)
- 5 × 2x = 10x
- 5 × −3 = −15
3. How do you expand double brackets?
To expand double brackets, multiply every term in the first bracket by every term in the second bracket. This is sometimes called the grid method or distributive method.
Example:
(x + 2)(x + 3)
- x × x = x²
- x × 3 = 3x
- 2 × x = 2x
- 2 × 3 = 6
x² + 3x + 2x + 6 = x² + 5x + 6.
4. What is the formula for expanding (a + b)²?
The formula for expanding (a + b)² is a² + 2ab + b². This is a standard algebraic identity.
It comes from:
(a + b)(a + b)
- a × a = a²
- a × b = ab
- b × a = ab
- b × b = b²
5. What is the formula for expanding (a − b)²?
The formula for expanding (a − b)² is a² − 2ab + b². This is another key algebraic identity.
It comes from:
(a − b)(a − b)
- a × a = a²
- a × −b = −ab
- −b × a = −ab
- −b × −b = b²
6. How do you expand brackets with negative signs?
When expanding brackets with a negative sign, multiply every term inside the bracket by the negative value. This changes the sign of each term.
Example:
−3(x − 4)
- −3 × x = −3x
- −3 × −4 = +12
7. What is the difference between expanding and factorising?
Expanding means multiplying out brackets, while factorising means writing an expression as a product of brackets. They are opposite processes in algebra.
Example:
- Expanding: 2(x + 5) = 2x + 10
- Factorising: 2x + 10 = 2(x + 5)
8. How do you expand three brackets?
To expand three brackets, expand two brackets first, then multiply the result by the third bracket.
Example:
(x + 1)(x + 2)(x + 3)
- First expand (x + 1)(x + 2) = x² + 3x + 2
- Then multiply (x² + 3x + 2)(x + 3)
9. Why do we combine like terms after expanding?
We combine like terms after expanding to simplify the expression into its final form. Like terms have the same variables raised to the same powers.
Example:
x² + 3x + 2x + 6
- 3x and 2x are like terms.
- 3x + 2x = 5x
10. What are common mistakes when expanding expressions?
Common mistakes when expanding expressions include missing terms, sign errors, and forgetting to multiply every term. Careful use of the distributive property helps avoid errors.
Typical mistakes:
- Not multiplying every term inside the bracket.
- Incorrect handling of negative signs.
- Forgetting to combine like terms.
