How to Factor Polynomials Using Common Methods
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 in Algebra Made Simple
1. What is factoring in mathematics?
Factoring is the process of writing an expression as a product of simpler expressions called factors. In algebra, factoring usually means rewriting a polynomial as multiplication instead of addition or subtraction.
- Example: x² − 5x + 6 can be factored as (x − 2)(x − 3).
- Factoring is the reverse process of expanding brackets.
- It is commonly used to solve quadratic equations and simplify algebraic expressions.
2. How do you factor a quadratic equation?
To factor a quadratic equation of the form ax² + bx + c, you find two numbers that multiply to ac and add to b. Follow these steps:
- Step 1: Multiply a × c.
- Step 2: Find two numbers that multiply to ac and add to b.
- Step 3: Rewrite the middle term and factor by grouping.
3. What is the greatest common factor (GCF)?
The greatest common factor (GCF) is the largest number or algebraic term that divides all terms in an expression exactly. To find the GCF:
- Find the common factors of the coefficients.
- Take the smallest power of common variables.
4. How do you factor by grouping?
Factoring by grouping means grouping terms in pairs and factoring out common factors from each pair. Steps:
- Step 1: Group the first two and last two terms.
- Step 2: Factor out the GCF from each group.
- Step 3: Factor out the common binomial.
5. What is the difference between factoring and expanding?
Factoring rewrites an expression as a product of factors, while expanding multiplies expressions to remove brackets. In simple terms:
- Factoring: x² − 9 → (x − 3)(x + 3)
- Expanding: (x − 3)(x + 3) → x² − 9
6. How do you factor a difference of squares?
A difference of squares follows the formula a² − b² = (a − b)(a + b). This applies when both terms are perfect squares and separated by subtraction.
- Example: x² − 16 = (x − 4)(x + 4)
- Example: 9y² − 25 = (3y − 5)(3y + 5)
7. What is factoring trinomials?
Factoring trinomials means rewriting a three-term polynomial, usually in the form x² + bx + c, as a product of two binomials. You find two numbers that:
- Multiply to c
- Add to b
8. Can you factor a polynomial with four terms?
Yes, a polynomial with four terms can often be factored using factoring by grouping. You typically:
- Group terms in pairs.
- Factor out common factors from each pair.
- Factor out the common binomial.
9. Why is factoring important in solving equations?
Factoring is important because it helps solve equations using the zero product property, which states that if ab = 0, then a = 0 or b = 0. Example:
- Solve x² − 5x + 6 = 0
- Factor → (x − 2)(x − 3) = 0
- Solutions: x = 2 or x = 3
10. What are common mistakes when factoring polynomials?
Common mistakes when factoring polynomials include missing common factors, sign errors, and incorrect number pairs. Watch out for:
- Not factoring out the GCF first.
- Mixing up positive and negative signs.
- Forgetting that a² + b² cannot be factored using real numbers.
- Not checking by expanding your final answer.
