Factoring is a process used in mathematics that consists of representing a number or an expression as a product of factors.
By writing a polynomial like the multiplication of other polynomials, we can often simplify the expression.
Check out the types of polynomial factorization below:
Common Factor in Evidence
We use this type of factorization when there is a factor that repeats itself in all terms of the polynomial.
This factor, which can contain numbers and letters, will be placed in front of the parentheses.
Inside the parentheses will be the result of dividing each term of the polynomial by the common factor.
In practice, let's do the following steps:
1º) Identify if there is a number that divides all the coefficients of the polynomial and letters that are repeated in all terms.
2º) Put the common factors (number and letters) in front of the parentheses (in evidence).
3rd) Put inside parentheses the result of dividing each factor of the polynomial by the factor that is in evidence. In the case of letters, we use the rule of division of powers of the same base.
Examples
a) What is the factored form of the polynomial 12x + 6y - 9z?
First, we identify that the number 3 divides all the coefficients and that there is no letter that repeats.
We put the number 3 in front of the parentheses, we divide all the terms by three and the result we will put inside the parentheses:
12x + 6y - 9z = 3 (4x + 2y - 3z)
b) Factor 2a2b + 3a3c - a4.
As there is no number that divides 2, 3 and 1 at the same time, we will not put any number in front of the parentheses.
The letter The is repeated in all terms. The common factor will be the The2, which is the smallest exponent of The in expression.
We divide each term of the polynomial by The2:
2nd2 b: the2 = 2nd2 - 2 b = 2b
3rd3c: the2 = 3rd3 - 2 c = 3ac
The4: a2 = the2
We put the The2 in front of parentheses and the results of divisions within parentheses:
2nd2b + 3a3c - a4 = the2 (2b + 3ac - a2)
grouping
In the polynomial that does not exist a factor that repeats itself in all terms, we can use the factorization by grouping.
For this, we must identify terms that can be grouped by common factors.
In this type of factorization, we put the common factors of the groupings in evidence.
Example
Factor the polynomial mx + 3nx + my + 3ny
The terms mx and 3nx has as a common factor the x. already the terms my and 3ny have as a common factor the y.
Putting these factors in evidence:
x (m + 3n) + y (m + 3n)
Note that (m + 3n) is now also repeated in both terms.
Putting it in evidence again, we find the factored shape of the polynomial:
mx + 3nx + my + 3ny = (m + 3n) (x + y)
Perfect Square Trinomial
Trinomials are polynomials with 3 terms.
The perfect square trinomials a2 + 2ab + b2 and the2 - 2ab + b2 result from the remarkable product of the type (a + b)2 and (a - b)2.
Thus, the factorization of the perfect square trinomial will be:
The2 + 2ab + b2 = (a + b)2 (square of the sum of two terms)
The2 - 2ab + b2 = (a - b)2 (square of the difference of two terms)
To find out if a trinomial really is a perfect square, we do the following:
1º) Calculate the square root of the terms that appear squared.
2) Multiply the values found by 2.
3rd) Compare the value found with the term that does not have squares. If they are equal, it's a perfect square.
Examples
a) Factor the polynomial x2 + 6x + 9
First, we have to test whether the polynomial is a perfect square.
√x2 = x and √9 = 3
Multiplying by 2, we find: 2. 3. x = 6x
Since the value found is equal to the term that is not squared, the polynomial is perfect squared.
Thus, the factorization will be:
x2 + 6x + 9 = (x + 3)2
b) Factor the polynomial x2 - 8xy + 9y2
Testing if it's a perfect square trinomial:
√x2 = x and √9y2 = 3y
Doing the multiplication: 2. x. 3y = 6xy
The value found does not match the term of the polynomial (8xy ≠ 6xy).
Since it's not a perfect square trinomial, we can't use this type of factorization.
Difference of two squares
To factor polynomials of type a2 - B2 we use the remarkable product of sum and difference.
Thus, the factorization of polynomials of this type will be:
The2 - B2 = (a + b). (a - b)
To factor, we must calculate the square root of the two terms.
Then write the product of the sum of the values found and the difference between these values.
Example
Factor the 9x binomial2 - 25.
First, find the square root of the terms:
√9x2 = 3x and √25 = 5
Write these values as a product of the sum and the difference:
9x2 - 25 = (3x + 5). (3x - 5)
perfect cube
the polynomials a3 + 3rd2b+3ab2 + b3 and the3 - 3rd2b+3ab2 - B3 result from the remarkable product of the type (a + b)3 or (a - b)3.
Thus, the factored shape of the perfect cube is:
The3 + 3rd2b+3ab2 + b3 = (a + b)3
The3 - 3rd2b+3ab2 - B3 = (a - b)3
To factor out polynomials of this type, we must calculate the cubic root of the terms to the cube.
Afterwards, it is necessary to confirm that the polynomial is a perfect cube.
If so, we cube the sum or subtraction of the values of the cubic roots found.
Examples
a) Factor the polynomial x3 + 6x2 + 12x + 8
First, let's calculate the cubic root of the terms cubed:
3√ x3 = x and 3√ 8 = 2
Then confirm if it's a perfect cube:
3. x2. 2 = 6x2
3. x. 22 = 12x
Since the terms found are the same as the terms in the polynomial, then it is a perfect cube.
Thus, the factorization will be:
x3 + 6x2 + 12x + 8 = (x + 2)3
b) Factor the polynomial a3 - 9th2 + 27th - 27th
First let's calculate the cubic root of the terms cubed:
3to3 = a and 3√ - 27 = - 3
Then confirm if it's a perfect cube:
3. The2. (-3) = - 9th2
3. The. (- 3)2 = 27th
Since the terms found are the same as the terms in the polynomial, then it is a perfect cube.
Thus, the factorization will be:
The3 - 9th2 + 27a - 27 = (a - 3)3
Read too:
- Potentiation
- Polynomials
- Polynomial Function
- Prime numbers
Solved Exercises
Factor the following polynomials:
a) 33x + 22y - 55z
b) 6nx - 6ny
c) 4x – 8c + mx – 2mc
d) 49 - the2
e) 9th2 + 12th + 4
a) 11. (3x + 2y - 5z)
b) 6n. (x - y)
c) (x – 2c). (4 + m)
d) (7 + a). (7 - a)
e) (3rd + 2)2
See too:
- Algebraic Expressions
- Exercises on Algebraic Expressions
- Notable products
- Notable Products - Exercises
