How to Solve Factorisation Problems Using Division
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
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2. Divide: 3x3 + 4x + 11 x3 - 3x + 2
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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 Division Made Easy
1. What is factorisation using the division method in algebra?
Factorisation using the division method is a process for breaking down a polynomial into a product of its factors. It involves dividing the polynomial (the dividend) by a smaller polynomial (the divisor). If the division results in a remainder of zero, it confirms that the divisor is a factor of the original polynomial. This method is especially useful for higher-degree polynomials.
2. Can you provide a simple example of factorising a polynomial using division?
Certainly. Let's factorise the polynomial p(x) = x² + 7x + 12. If we suspect that (x + 3) is a factor, we can use long division.
1. Divide the first term of the dividend (x²) by the first term of the divisor (x), which gives x. This is the first term of the quotient.
2. Multiply the divisor (x + 3) by x to get x² + 3x, and subtract this from the dividend. We are left with 4x + 12.
3. Divide the first term of the new dividend (4x) by x, which gives 4. This is the second term of the quotient.
4. Multiply the divisor (x + 3) by 4 to get 4x + 12, and subtract. The remainder is 0.
Since the remainder is zero, (x + 3) is a factor, and the quotient (x + 4) is the other factor. Thus, x² + 7x + 12 = (x + 3)(x + 4).
3. How is using division for factorisation different from regular polynomial division?
The primary difference lies in the goal of the process. In regular polynomial division, the objective is to find the quotient and the remainder. In factorisation, division is used as a test; the main goal is to check if the remainder is zero. A zero remainder proves that the divisor is a factor, allowing you to express the original polynomial as a product of simpler factors.
4. How do you know which polynomial to use as the divisor when trying to factorise a larger polynomial?
This is a key challenge that can be solved using the Factor Theorem concept. You can test simple integer values (like 1, -1, 2, -2, etc.) in the polynomial. If you find a value 'a' such that the polynomial evaluates to zero (i.e., p(a) = 0), then (x - a) is a factor. You can then use (x - a) as your divisor for the long division method to find the other factors.
5. What does it mean if the remainder is not zero when using the division method for factorisation?
If the division process results in a non-zero remainder, it simply means that the polynomial you used as the divisor is not a factor of the dividend polynomial. The factorisation attempt with that particular divisor has failed, and you would need to try a different potential factor.
6. When is it better to use the division method for factorisation over other methods like grouping or algebraic identities?
The division method is most powerful and often necessary for higher-degree polynomials, such as cubic or quartic expressions, where the factors are not immediately obvious. While methods like common factorisation, regrouping, and standard identities (e.g., a² - b²) are faster for simpler polynomials, the division method provides a systematic approach when these simpler techniques do not apply.
7. Does the factorisation by division method apply to both monomial and polynomial divisors?
Yes, the principle of division applies to both, but the technique differs slightly:
- Dividing by a Monomial: When dividing a polynomial by a single-term expression (a monomial), you can divide each term of the polynomial individually by that monomial. For example, (8x³ + 4x²) ÷ (2x) = (8x³ ÷ 2x) + (4x² ÷ 2x) = 4x² + 2x.
- Dividing by a Polynomial: When the divisor has two or more terms, you must use the long division method to systematically find the quotient and check for a zero remainder.
