Factoring or factor decomposition is the process of presenting a mathematical expression or a number in multiplication form . Remember that the factors are the elements of multiplication and the result is known as the product .
Types of factoring
In general terms, we can talk about two types of factorization: the factorization of integers and the factorization of algebraic expressions.
Every whole number can be decomposed into its prime factors . A prime number is one that is only divisible between 1 and itself. For example, the 2 can only be divided by 1 and 2.
We can decompose a given number X as the multiplication of its prime factors. For example, the number 525 is made up of the prime numbers 5, 3, and 7 as follows:
Factoring algebraic expressions
The goal of factoring is to take a complicated polynomial and express it as the product of its simple polynomial factors.
Are called factors or dividers of an algebraic expression to algebraic expressions multiplied each given as the first expression product. For instance:
The factors are:
How to factor
When we talk about factoring, we can follow the following recommendations:
- Observe if there is a common factor, that is, if there is a factor that is repeated in the different terms.
- Order the expression: sometimes when arranging the expression we realize the possibilities of factoring.
- Find out if the expression is factorizable: sometimes we are in the presence of expressions that cannot be decomposed into factors.
- Check if the factors found are in turn factorable.
Steps to find the common factor of a polynomial
The common factor of a polynomial is the step prior to factoring a polynomial. We are going to explain step by step how to find the common factor of the following polynomial:
We get the greatest common factor of 24 and 16. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24; the factors of 16 are 1, 2, 4, 8, and 16. The greatest common factor is 8.
We get the common factors of the variables, in this case the common variables with the highest common power. The common variables are x and y . The greatest common power of x is x 6 and the greatest common power of y is y 3 .
We write the common factor of the polynomial as the product of steps 1 and 2 above:
Factorization of polynomials
We already know the common factor of the polynomial, so we can go on to factor:
We determine the common factor of the polynomial:
We rewrite each term of the polynomial as a function of the common factor. For this, we first divide the term by the common factor to obtain a second factor:
Now we substitute each term for the common factor and the respective second factor:
Note : 8x 6 y 3 (3x 2 ) – 8x 6 y 3 (2y 4 z 3 ) is not the factored form because the factors are not separated yet.
We use the distributive property to get the common factor:
We review the steps taken:
We can factor a polynomial of four terms by grouping them into pairs. Let’s look at the following example:
We rearrange the terms such that the first two have a common factor and the other two also have a common factor:
We factor the x of the first term and the y as a common factor of the second term:
We use the distributive property to factor the term (a + b) from the expression:
Factoring a quadratic equation
When we have a three-term polynomial, this can be a quadratic trinomial of the form ax 2 + bx + c . This expression is obtained from the multiplication of two binomials:
When factoring a quadratic equation like x 2 + 9x + 14 , we want to get the two binomials that originated it: (x + 7) (x + 2) .
Factor a quadratic equation by trial and error
For the expression 4x 2 -11x-3 we look for two binomial factors. 4x 2 is the first term, so the multiplication of the first numerical coefficients of the binomials must be 4. The last term is -3, so the last terms of the factors have different signs whose product is -3. We can try various combinations:
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Factor a quadratic equation by grouping
To factor by grouping, we identify the coefficients a , b, and c and look for two factors ac whose sum is b . For example, for the equation 4x 2 -11x-3 , the coefficients are a = 4, b = -11, and c = -3.
The factors ac = (4) (- 3) = – 12. Two factors of -12 that together give -11 are -12 and 1.
Now we replace the middle term of 4x 2 -11x-3 with -12x + 1x .
We group the terms in pairs and look for the common factor:
We apply the distributive property to the factor (x-3):
The factored form is then as:
Factorization of perfect square trinomials
A perfect square trinomial is one where the absolute value of the coefficient b is equal to twice the product of the roots of a and c :
For example, in the equation 4x 2 -20x + 25 , a = 4, b = -20, c = 25, then:
This indicates that 4x 2 -20x + 25 can be factored as the square of a binomial:
The first term will be the square root of 4x 2 and the last term is the square root of c :
The sign in the binomial is the same as the middle term of the trinomial.
See also Quadratic Equations of the Second Degree .
Factoring of binomials
The factorizable binomials are:
- the difference of two squares (x 2-y 2 ),
- the difference of two cubes (x 3-y 3 ) and
- the sum of two cubes (x 3+ y 3 ).
The factorization of the difference of two squares (x 2 -y 2 ) is:
The factorization of the difference of two cubes (x 3 -y 3 ) is:
The factorization of the sum of two cubes (x 3 + y 3 ) is:
- Notable products .
- Laws of exponents
Solved factoring exercises
1. Factor the following expression:
The common factor is (x-1).
Apply the distributive property to the factor (x-1):
2. Factor the following expression:
We take as a common factor 25x 2 and 2 z
We factor the difference of two squares which is 4x 2 -1.
The full factored form is:
3. Factor the following expression:
This expression is a quadratic equation, so we look for binomial factors:
We are looking for two numbers that multiplied to -30 and added to -7. We test with -10 and 3:
The factored form is:
4. Factor the following expression:
We note that this expression has four terms. We group them in pairs so that we can get a common factor:
We factor the square binomial (x 2 -1):
The final factored form is: