Introduction to Factoring
The process of finding factors is known as factoring. This process can also be called factorization. It can also be defined as, factoring consists of a number or any other mathematical object as the product of two or more factors. For example, 3 and 5 are the factors of integer 15.
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Factoring Algebra
Factoring algebra is the process of factoring algebraic terms. To understand it in a simple way, it is like splitting an expression into a multiplication of simpler expressions known as factoring expression example: 2y + 6 = 2(y + 3). Factoring can be understood as the opposite to the expanding. Different types of factoring algebra are given below so that you can learn about factoring in brief.
Types of Factoring Algebra
Different types of factoring algebra are discussed below:
Factoring out the Greatest Common factor.
The sum-product pattern.
The grouping pattern.
Perfect square trinomials.
Difference of Squares.
Let us discuss the basic two methods of factoring which is used frequently to factorise the polynomial. The most popular formula used to find the factors of a polynomial in the Quadratic equation is Shridhar’s formula.
\[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\]
Greatest Common Factor
In this, we have to find the greatest common factor of the given polynomial to factorise it. This process is a type of reverse procedure of distributive law.
Distributive law p(q + r) = pq + pr
In the case of factorisation, it is opposite of distributive law
pq + pr = p(q + r)
Where ‘p’ is the greatest common factor of the given polynomial.
Factorisation Problems:
1) Factorise 6x2 + 3x
Now, take the common multiple out from the above polynomial
3 is the common multiple for the given problem.
= 3x (2x + 1)
Therefore, 6x2 + 3x = 3x(2x + 1)
2) Factorise 2x2 + 8x
Here, 2 is the common multiple for the given problem
= 2x(x + 4)
Therefor, 2x2 + 8x = 2x(x + 4)
Note: This method is applicable if each term in the polynomial shares a common factor.
Factoring Polynomial by Grouping
In this method, the given polynomial is grouped in the pairs to find the zeros. This method is also called factoring by pairs.
Factorisation examples: Factorise x2 - 15x + 50
To solve the problem by grouping, find the two numbers which when added gives -15 and multiplication gives 50.
So, -5 and -10 are two numbers
Like, (-5) × (-10) = 50
(-5) + (-10) = -15
Therefore, The given polynomial can be written as:
X2 - 5x - 10x + 50 = 0
x(x - 5) - 10(x - 5) = 0
In this taking (x - 5) as common factor;
We get, (x - 5)(x - 10)
Hence, The factors are (x - 5) and (x - 10).
Note: This method is applicable if the polynomial of the form x2 + bx + c and there are factors of ac that add up to b.
Factoring Rules:
Some of the basic rules of the factoring are:
If the third co-efficient(c) is "plus", then the factors will be either both "plus" or else both "minus".
If the second coefficient(b) is "plus", then the factors are both "plus".
If the second coefficient(b) is "minus", then the factors are both "minus".
In either case, look for factors that add to b.
If the third coefficient(c) is "minus", then the factors will be of alternating signs; that is, one will be "plus" and one will be "minus".
If the second coefficient(b) is "plus", then the larger of the two factors is "plus".
If the second coefficient(b) is "minus", then the larger of the two factors is "minus".
FAQs on Factoring
1. What is meant by factoring in Maths, with an example?
In mathematics, factoring is the process of breaking down an algebraic expression into a product of simpler expressions, known as its factors. When these factors are multiplied together, they result in the original expression. For example, the expression x² - 9 can be factored into (x - 3)(x + 3), because multiplying these two factors gives you the original expression back.
2. What are the main methods for factoring algebraic expressions as per the CBSE syllabus?
According to the CBSE curriculum for middle school, there are several key methods for factoring algebraic expressions:
- Taking out the Greatest Common Factor (GCF): Finding the largest factor common to all terms in the expression.
- Factoring by Grouping: Grouping terms with common factors when there isn't a single GCF for all terms.
- Using Algebraic Identities: Applying standard formulas like the difference of squares (a² - b²) or perfect square trinomials ((a+b)² or (a-b)²).
- Splitting the Middle Term: A method used for factoring quadratic trinomials of the form ax² + bx + c.
3. How are algebraic identities like a² - b² used for factoring?
Algebraic identities are powerful shortcuts for factoring. The difference of squares identity, a² - b² = (a - b)(a + b), is used when an expression consists of one squared term being subtracted from another. To use it, you identify the 'a' and 'b' terms by taking the square root of each part of the expression. For instance, to factor 49x² - 16, you identify a = 7x and b = 4, so the factors are (7x - 4)(7x + 4).
4. What is the process for factoring a quadratic trinomial by splitting the middle term?
This method is used for trinomials like x² + 7x + 12. The process is:
- Step 1: Find two numbers that multiply to the constant term (12) and add up to the coefficient of the middle term (7). In this case, the numbers are 3 and 4 (since 3 × 4 = 12 and 3 + 4 = 7).
- Step 2: Rewrite or 'split' the middle term using these two numbers: x² + 3x + 4x + 12.
- Step 3: Factor the new four-term expression by grouping. Group the first two terms and the last two terms: (x² + 3x) + (4x + 12).
- Step 4: Factor out the GCF from each group: x(x + 3) + 4(x + 3).
- Step 5: The final factors are (x + 4)(x + 3).
5. Why is finding the Greatest Common Factor (GCF) the most important first step in factoring?
Finding the Greatest Common Factor (GCF) should always be the first step because it simplifies the polynomial. By factoring out the GCF, the remaining expression becomes smaller and less complex. This makes it much easier to apply other factoring techniques like grouping or using identities. Skipping this step can lead to incomplete factorisation or make the problem much harder to solve.
6. What is the key difference between factoring an expression and solving an equation?
The key difference lies in the goal. Factoring an expression involves rewriting it as a product of its factors (e.g., turning x² - 25 into (x-5)(x+5)). There is no single 'answer'. In contrast, solving an equation means finding the specific numerical value(s) for the variable that make the equation true (e.g., for x² - 25 = 0, the solutions are x = 5 and x = -5). Factoring is often a method used to help solve an equation.
7. Where can the concept of factoring be applied in real-life situations?
While it seems abstract, the logic of factoring is used in many real-world scenarios. For example:
- Area and Dimensions: If you know the area of a rectangular space (e.g., 50 sq. ft.) and want to find possible dimensions, you are finding the factors of 50 (e.g., 5x10, 2x25).
- Event Planning: When arranging chairs or tables in equal rows, you are using factors to divide a total number into even groups.
- Cryptography: Advanced factoring is a core principle in creating secure codes for online banking and data protection.
8. Can every algebraic expression be factored?
No, not every algebraic expression can be factored using integers. An expression that cannot be factored into simpler expressions with integer coefficients is called a prime polynomial. This is conceptually similar to a prime number, which only has factors of 1 and itself. For example, the expression x² + 4 is considered prime over real numbers because it cannot be broken down further using this number system.
