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The process of writing a number or expression as a product is called factoringThe process of writing a number or expression as a product.. If we write the monomial , we say that the product is a factorizationAny combination of factors, multiplied together, resulting in the product. of and that and are factorsAny of the numbers or expressions that form a product.. Typically, there are many ways to factor a monomial. Some factorizations of follow:
Given two or more monomials, it will be useful to find the greatest common monomial factor (GCF)The product of the common variable factors and the GCF of the coefficients. of each. The GCF of the monomials is the product of the common variable factors and the GCF of the coefficients.
Find the GCF of and
Solution:
Begin by finding the GCF of the coefficients. In this case, and It should be clear that
Next determine the common variable factors with the smallest exponents.
The common variable factors are , , and Therefore, given the two monomials,
Answer:
It is worth pointing out that the GCF divides both expressions evenly.
Furthermore, we can write the following:
The factors and share no common monomial factors other than 1; they are relatively primeExpressions that share no common factors other than 1..
Determine the GCF of the following three expressions: , , and
Solution:
Begin by finding the GCF of the coefficients. To do this, determine the prime factorization of each and then multiply the common factors with the smallest exponents.
Therefore, the GCF of the coefficients of the three monomials is
Next, determine the common factors of the variables.
The variable factors in common are , , and Therefore,
Note that the variable c is not common to all three expressions and thus is not included in the GCF.
Answer:
The application of the distributive property is the key to multiplying polynomials. For example,
The process of factoring a polynomial involves applying the distributive property in reverse to write each polynomial as a product of polynomial factors.
Consider factoring the result of the opening example:
We see that the distributive property allows us to write the polynomial as a product of the two factors and Note that in this case, is the GCF of the terms of the polynomial.
Factoring out the greatest common factor (GCF)The process of rewriting a polynomial as a product using the GCF of all of its terms. of a polynomial involves rewriting it as a product where a factor is the GCF of all of its terms.
To factor out the GCF of a polynomial, we first determine the GCF of all of its terms. Then we can divide each term of the polynomial by this factor as a means to determine the remaining factor after applying the distributive property in reverse.
Factor out the GCF:
Solution:
In this case, the GCF(18, 30, 6) = 6, and the common variable factor with the smallest exponent is The GCF of the polynomial is
The missing factor can be found by dividing each term of the original expression by the GCF.
Apply the distributive property (in reverse) using the terms found in the previous step.
If the GCF is the same as one of the terms, then, after the GCF is factored out, a constant term 1 will remain. The importance of remembering the constant term becomes clear when performing the check using the distributive property.
Answer:
Factor out the GCF:
Solution:
The GCF of the terms is The last term does not have a variable factor of z, and thus z cannot be a part of the greatest common factor. If we divide each term by , we obtain
and can write
Answer:
Of course, not every polynomial with integer coefficients can be factored as a product of polynomials with integer coefficients other than 1 and itself. If this is the case, then we say that it is a prime polynomialA polynomial with integer coefficients that cannot be factored as a product of polynomials with integer coefficients other than 1 and itself.. For example, a linear factor such as is prime. However, it can be factored as follows:
If an x is factored out, the resulting factor is not a polynomial. If any constant is factored out, the resulting polynomial factor will not have integer coefficients. Furthermore, some linear factors are not prime. For example,
In general, any linear factor of the form , where a and b are relatively prime integers, is prime.
In this section, we outline a technique for factoring polynomials with four terms. First, review a preliminary example where the terms have a common binomial factor.
Factor:
Solution:
Begin by rewriting the second term as Next, consider as a common binomial factor and factor it out as follows:
Answer:
Factoring by groupingA technique for factoring polynomials with four terms. is a technique that enables us to factor polynomials with four terms into a product of binomials. This involves an intermediate step where a common binomial factor will be factored out. For example, we wish to factor
Begin by grouping the first two terms and the last two terms. Then factor out the GCF of each grouping:
In this form, the polynomial is a binomial with a common binomial factor,
Therefore,
We can check by multiplying.
Factor by grouping:
Solution:
The GCF for the first group is We have to choose 5 or −5 to factor out of the second group.
Factoring out +5 does not result in a common binomial factor. If we choose to factor out −5, then we obtain a common binomial factor and can proceed. Note that when factoring out a negative number, we change the signs of the factored terms.
Answer: Check by multiplying; this is left to the reader as an exercise.
Sometimes we must first rearrange the terms in order to obtain a common factor.
Factor:
Solution:
Simply factoring the GCF out of the first group and last group does not yield a common binomial factor.
We must rearrange the terms, searching for a grouping that produces a common factor. In this example, we have a workable grouping if we switch the terms and
Answer:
Not all factorable four-term polynomials can be factored with this technique. For example,
This four-term polynomial cannot be grouped in any way to produce a common binomial factor. Despite this, the polynomial is not prime and can be written as a product of polynomials. It can be factored as follows:
Factoring such polynomials is something that we will learn to do as we move further along in our study of algebra. For now, we will limit our attempt to factor four-term polynomials to using the factor by grouping technique.
A binomial is a polynomial with two terms. We begin with the special binomial called difference of squares where a and b represent algebraic expressions.:
To verify the above formula, multiply.
We use this formula to factor certain special binomials.
Factor:
Solution:
Identify the binomial as difference of squares and determine the square factors of each term.
Here we can write
Substitute into the difference of squares formula where and
Multiply to check.
Answer:
Factor:
Solution:
First, identify this expression as a difference of squares.
Use and in the formula for a difference of squares and then simplify.
Answer:
Given any real number b, a polynomial of the form is prime. Furthermore, the sum of squares where a and b represent algebraic expressions. This does not have a general factored equivalent. does not have a general factored equivalent. Care should be taken not to confuse this with a perfect square trinomial.
Therefore,
For example, the sum of squares binomial is prime. Two other special binomials of interest are the sum, where a and b represent algebraic expressions. and difference of cubes, where a and b represent algebraic expressions.:
We can verify these formulas by multiplying.
The process for factoring sums and differences of cubes is very similar to that of differences of squares. We first identify a and b and then substitute into the appropriate formula. The separate formulas for the sum and difference of cubes allow us to always choose a and b to be positive.
Factor:
Solution:
First, identify this binomial as a difference of cubes.
Next, identify what is being cubed.
In this case, and Substitute into the difference of cubes formula.
We can check this factorization by multiplying.
Answer:
It may be the case that the terms of the binomial have a common factor. If so, it will be difficult to identify it as a special binomial until we first factor out the GCF.
Factor:
Solution:
The terms are not perfect squares or perfect cubes. However, notice that they do have a common factor. First, factor out the GCF,
The resulting binomial factor is a sum of cubes with and
Answer:
When the degree of the special binomial is greater than two, we may need to apply the formulas multiple times to obtain a complete factorization. A polynomial is completely factoredA polynomial that is prime or written as a product of prime polynomials. when it is prime or is written as a product of prime polynomials.
Factor completely:
Solution:
First, identify what is being squared.
To do this, recall the power rule for exponents, When exponents are raised to a power, multiply them. With this in mind, we find
Therefore, and Substitute into the formula for difference of squares.
At this point, notice that the factor is itself a difference of two squares and thus can be further factored using and The factor is prime and cannot be factored using real numbers.
Answer:
When factoring, always look for resulting factors to factor further.
Factor completely:
Solution:
This binomial is both a difference of squares and difference of cubes.
When confronted with a binomial that is a difference of both squares and cubes, as this is, make it a rule to factor using difference of squares first. Therefore, and Substitute into the difference of squares formula.
The resulting two binomial factors are sum and difference of cubes. Each can be factored further. Therefore, we have
The trinomial factors are prime and the expression is completely factored.
Answer:
As an exercise, factor the previous example as a difference of cubes first and then compare the results. Why do you think we make it a rule to factor using difference of squares first?
Determine the GCF of the given expressions.
, ,
, ,
, ,
, ,
, ,
, ,
, ,
, ,
, ,
, ,
Determine the missing factor.
Factor out the GCF.
Factor by grouping.
Factor.
Given , show that
Given , show that
Given , show that
Given , show that
Given , show that
Given , show that
What can be said about the degree of a factor of a polynomial? Give an example.
If a binomial falls into both categories, difference of squares and difference of cubes, which would be best to use for factoring, and why? Create an example that illustrates this situation and factor it using both formulas.
Write your own examples for each of the three special types of binomial. Factor them and share your results.
Answer may vary
Answer may vary
Answer may vary
Answer may vary
Answer may vary