How to Do Factorisation Using Division with Steps and Examples
What is Factorization?
Let us first learn what is factorization? Factorization is nothing but the process of minimizing the bracket of a quadratic equation, rather than expanding the bracket and transforming the equation to a product of factors which cannot be reduced further. For example, factorising (x²+ 5x+ 6) brings forth the result as (x+2) (x+3). Here, (x+2) (x+3) denotes a factorization of a polynomial (x²+5x+6). These factors can either be algebraic expressions, variables, or integers. Basically, factorisation is the reverse function of multiplication. A form of decomposition,factorization brings forth the gradual splitting of a polynomial into its factors. In this article, we will discuss what is factorization, factorization using division, factorization by division, finding factors by long division,etc.
Factorization Definition
Factorisation or factoring is defined as the splitting or disintegration of a unit (for instance a number, a matrix, or a polynomial) into a product of another unit or factors, which when multiplied together obtains the original number or a matrix, etc.
It is simply the transformation of an integer or polynomial into factors such that when two integers multiplied together obtains the result as the original integer or polynomial. In the factorization method, we minimise any algebraic or quadratic equation into its simpler form, where the equations are expressed as the product of factors rather than expanding the brackets. The factors of any equation can either be an integer, a variable, or an algebraic expression itself
Factorization Using Division Important Points
Following points should be noted while factoring the polynomial using the division method.
Factorization by division method is a simpler and traditional approach of finding the factors of a polynomial expression.
Determining factors of a polynomial is simply like performing any simple division, the only thing should be considered is the accuracy of variables and coefficients.
There are two ways through which factorization of polynomials can be done. The first method is the simple division method and the second method is the long division method.
Finding Factors: The long Division Way Introduction
The following are the steps to be followed for finding factors by the long division way.
We will first arrange the given polynomials in descending order. We will replace every missing term with 0.
In the second step, we will divide the first term of the dividend by the first term of the divisor. With this, we will get the first term of the quotient.
Next, we have to multiply the divisor by the first term of the quotient.
Further, the next term will be brought down by subtracting the product from the dividend. The next term which is brought down and the difference of the product and dividend will be the new dividend.
Repeat step 2 and 4 to determine the second term of the quotient.
Continue the process till we get a reminder. The remainder can be either zero or lower than the divisor.
We will get the value of the remainder equals to zero if the divisor is a factor of dividend. We will get the remainder lower than the divisor if the value of the remainder is not equal to zero.
Factorization By Division
The first step is to split the polynomials into its direct factors in the factorization by simple division method. For example if we divide, 8z³ + 7z² +6z by 2, we will split the given equation in its basic factors i.e.2x(4z)² + 2x(7/2 × z) + 2z(3).
In the next step, we will write the common factors separately i.e. 2z{(4z)² +(7/2 × z) + (3)}/2z.
In the last step, we will divide the given expression as mentioned in the question i.e 2z{(4z)² +(7/2 × z) + (3)}.
Hence, the answer will be: 4z² +(7/2z) + 3.
Factorization By Division Example
Factorise the below by simple division method:
Divide: (p2qr+ pq2r+ pqr2) by 4pqr
Solution: 2 ×2× 2× 2 [( p × p × r × r) + ( p × q× q ÷ r) +( p ×q × r × r)]
16pqr (p + q+ r)
Now, we will divide the polynomial as asked in the question
= 4×4 pqr (p + q+ r)/ 4pqr
= 4(p+q+r)
Solved Examples
1. Factorise the Following By Division Method
Divide: 3x3 + 4x + 11 x2 - 3x + 2
[Image will be Uploaded Soon]
2. Divide: 3x3 + 4x + 11 x3 - 3x + 2
[Image will be Uploaded Soon]
Step 11 : 3x² - 5x + 6
QuizTime
1. Factorization of 4x-20 Gives
2x-4
4(x-5)
5(x-4)
20x
2. Factorization of p+ pq + 2q+ 2q²
p² + 2pq
p+ q
(1+q)(p+2q)
None of the above
FAQs on Factorisation Using the Division Method in Algebra
1. What is factorisation using division?
Factorisation using division is a method of breaking a polynomial into factors by dividing it by a known factor. It is commonly used when one factor is already given or found using the Factor Theorem.
- Divide the polynomial by a suspected factor.
- If the remainder is 0, the divisor is a factor.
- Write the polynomial as the product of the divisor and the quotient.
2. How do you factorise a polynomial using long division?
To factorise a polynomial using long division, divide the polynomial by a known factor and express it as divisor × quotient. Follow these steps:
- Arrange terms in descending powers.
- Divide the first term of the dividend by the first term of the divisor.
- Multiply and subtract.
- Repeat until remainder is obtained.
3. How do you use synthetic division in factorisation?
Synthetic division is a shortcut method used to divide a polynomial by a linear factor of the form (x − a). Steps include:
- Write coefficients of the polynomial.
- Use the zero value a.
- Perform multiply-add steps.
4. What is the Factor Theorem in division method?
The Factor Theorem states that if f(a) = 0, then (x − a) is a factor of the polynomial f(x). This means:
- Substitute a into the polynomial.
- If the result is 0, division by (x − a) will give remainder 0.
5. Can you give an example of factorisation using division?
Yes, for example, factorise x² − 5x + 6 using division. Since f(2) = 0, (x − 2) is a factor.
- Divide x² − 5x + 6 by (x − 2).
- The quotient is x − 3.
6. When should you use division for factorisation?
Division is used for factorisation when a factor is known or suspected, especially in higher-degree polynomials. It is helpful when:
- A root is found using the Factor Theorem.
- The polynomial is cubic or higher.
- Other simple methods do not work easily.
7. What happens if the remainder is not zero in polynomial division?
If the remainder is not zero, then the divisor is not a factor of the polynomial. According to the Remainder Theorem:
- The remainder equals f(a) when dividing by (x − a).
- A non-zero remainder means no exact factorisation.
8. What is the difference between long division and synthetic division?
Long division works for all polynomial divisors, while synthetic division only works for linear divisors of the form (x − a).
- Long division: Detailed and works for any polynomial divisor.
- Synthetic division: Faster but limited to linear factors.
9. How do you factorise a cubic polynomial using division?
To factorise a cubic polynomial, first find one root and then divide to reduce it to a quadratic. Steps:
- Use trial values or Factor Theorem to find a root.
- Divide by (x − a).
- Factorise the resulting quadratic.
10. What are common mistakes in factorisation using division?
Common mistakes include sign errors and incorrect subtraction during division. Watch out for:
- Incorrect arrangement of terms in descending powers.
- Forgetting zero coefficients for missing terms.
- Arithmetic mistakes during multiply-subtract steps.
