Algebra is the branch of maths that deals with symbols and the rules for manipulating these symbols. In this section, we'll delve deeper into the foundational techniques of simplifying expressions and factorising, ensuring that you have a robust understanding of these essential skills.
Simplifying Expressions
Simplifying an expression means to make it as concise as possible. This involves combining like terms, using the distributive property, and understanding the properties of numbers.
Combining Like Terms
Like terms are terms that have the same variables raised to the same power.
Example 1:
Expression: 3x + 5x
Solution: Combine like terms. 3x + 5x = 8x
Example 2:
Expression: 4y2 - 2y2 + 7
Solution: Combine like terms. 4y2 - 2y2 = 2y2 So, the simplified expression is 2y2 + 7.
Distributive Property
The distributive property allows you to multiply a number outside the parenthesis by each term inside the parenthesis.
Example 3:
Expression: 2a(3a + 5)
Solution: Distribute the 2a. 2a * 3a = 6a2 2a * 5 = 10a So, the simplified expression is 6a2 + 10a.
Factorising Techniques
Factorising is the process of breaking down an expression into its simplest factors. It's the reverse process of expanding. Let's explore some common factorising techniques:
Common Factor
When two or more terms have a common factor, we can take it out.
Example 4:
Expression: 6x2 + 9x
Solution: The common factor is 3x. Factoring out 3x, we get: 3x(2x + 3)
Difference of Two Squares
This is a special factorisation technique used when there's a difference between two squared terms.
Example 5:
Expression: a2 - b2
Solution: This expression a2 - b2 is a difference of squares. The difference of squares formula is: a2-b2 = ( a + b) ( a - b). Applying this formula to the given expression: Therefore, this can be factorised as: (a + b)(a - b).
Trinomial Factorisation
When an expression has three terms, it might be factorisable into two binomial expressions.
Example 6:
Expression: x2 + 5x + 6
Solution: To factor the quadratic expression x2+5x+6, we can look for two numbers that multiply to give 6 (the constant term) and add up to 5 (the coefficient of the linear term).The numbers that fit these criteria are 2 and 3 because 2 x3 = 6 and 2 + 3 = 5. Therefore, this can be factorised as: (x + 2)(x + 3).
Quadratic Factorisation
For a quadratic expression of the form ax2 + bx + c, we look for two numbers that multiply to give ac and add to give b.
Example 7:
Expression: 6x2 + 5x - 6
Solution: Here, ac = -36. The numbers that multiply to give -36 and add to give 5 are 9 and -4. Rewrite the middle term using these numbers: 6x2 + 9x - 4x - 6 Group the terms: 3x(2x + 3) - 2(2x + 3) Factorise: (3x - 2)(2x + 3)
Perfect Square Trinomials
These are trinomials that are the square of binomials.
Example 8:
Expression: x2 + 8x + 16
Solution:The first term, x2, is the square of x. The third term, 16, is the square of 4. The middle term, 8x, is twice the product of x and 4. Therefore, this can be factorised as: (x + 4)(x + 4) or (x + 4)2
Tips for Students:
- Always look for the greatest common factor first.
- Remember the special factorisation forms like the difference of squares and perfect square trinomials.
- For quadratic factorisation, practice is key. The more you practice, the quicker you'll identify the numbers to split the middle term.
Practice Questions:
1. Simplify the expression: 7z + 2z - 3
2. Factorise the expression: 4y2 - 16
3. Factorise the expression: 3m2 + 12m
4. Simplify the expression: 5(2x + 4)
5. Factorise the expression: x2 + 8x + 16
Answers:
1. 9z - 3
2. 2y(2y - 4) or 2y(2y + 4)
3. 3m(m + 4)
4. 10x + 20
5. (x + 4)(x + 4) or (x + 4)2
FAQ
While understanding the foundational principles is paramount, there are some shortcuts that experienced mathematicians use. For instance, when multiplying binomials, the FOIL (First, Outer, Inner, Last) method can speed up the process. Recognising common factorised forms, like the difference of squares, can also save time. However, it's essential to use these shortcuts judiciously. Relying too heavily on them without a solid understanding can lead to mistakes. It's always a good idea to double-check your work, especially when using shortcuts, to ensure accuracy.
Choosing the right factorising technique often depends on recognising patterns in the expression you're working with. Start by looking for a common factor among all terms; if one exists, factor it out. If the expression is a trinomial, it might be factorisable into two binomials, especially if it resembles a quadratic form. Recognising special forms, like the difference of squares or perfect square trinomials, can also guide your choice. With practice, you'll become more adept at quickly identifying the structure of an expression and the most suitable factorising technique to apply.
Terms can only be combined (or "collected") when they are "like terms", meaning they have the same variables raised to the same powers. For example, 3x and 5x are like terms because they both have the variable x raised to the power of 1. However, 3x and 5x2 are not like terms because one has x raised to the power of 1, and the other has x raised to the power of 2. Trying to combine unlike terms would be akin to adding apples and oranges; they're fundamentally different. Always ensure that you're only combining like terms to maintain the integrity of the expression.
Factorising and expanding are two fundamental processes in algebra, but they are essentially opposite operations. Factorising involves expressing an algebraic expression as a product of its simplest factors. For instance, taking a quadratic expression and expressing it as a product of two binomials. On the other hand, expanding means multiplying out to remove parentheses or brackets, turning a product of terms into a sum. For example, multiplying the terms in a binomial product to get a quadratic expression. Understanding both processes is vital as they frequently complement each other in various mathematical problems and proofs.
Simplifying algebraic expressions is crucial for several reasons. Firstly, it helps in making complex problems more manageable and easier to understand. A simplified expression can often reveal patterns or characteristics that might not be immediately evident in its more complicated form. Secondly, in mathematical proofs or when solving equations, working with a more straightforward expression can prevent mistakes and make the process more efficient. Additionally, when comparing or combining different algebraic terms, having them in their simplest form ensures accuracy. Lastly, simplifying is a foundational skill in maths, and mastering it is essential for tackling more advanced algebraic concepts and techniques.
Practice Questions
To simplify the expression, we'll combine like terms. Starting with the terms containing x2: 4x2 - 3x2 = x2 Next, we'll combine the terms containing x: 7x + 2x = 9x So, the simplified expression is: x2 + 9x - 5
To factorise the expression, we'll look for the greatest common factor. In this case, the common factor is 2y. Taking out the common factor, we get: 2y(y + 4) So, the factorised form of the expression is 2y(y + 4).
