Factoring and the Distributive Property

 

Learning Objective(s)

·         Use the distributive property to factor a monomial out of a polynomial.

 

 

Factors are numbers that multiply together to produce another number. For example, 2 and 10 are factors of 20, as are 4 and 5 and 1 and 20. Factoring is the process of breaking a number down into its multiplicative factors. Factoring has many applications in mathematics; breaking a value down into smaller pieces gives us new ways to understand and manipulate it.

 

Just as any integer can be written as the product of factors, so too can any monomial or polynomial be expressed as a product of factors. Factoring is very helpful in simplifying and solving equations using polynomials.

 

The factors of a value always include what are called prime factors, which have no factors but 1 and themselves (in other words, they are prime numbers).  For example, 2 is a prime factor of 20 because its only factors are 1 and 2. But 10 is not a prime factor because 10 can be factored into not just 1 and 10 but also 2 and 5.

 

The process of breaking a number down into its prime factors is called prime factorization.

 

 

Let’s use a common factoring task—finding the greatest common factor (GCF) of two whole numbers—as a jumping off point. The GCF of two numbers is the largest number that is a factor of both of the numbers. To find the GCF, factor both numbers to find their prime factors, identify the prime factors they have in common, and then multiply those together.

 

 

Because the GCF is the product of the prime factors that these numbers have in common, we know that it is a factor of both numbers. (If you want to test this, go ahead and divide both 210 and 168 by 42—they are both evenly divisible by this number!)

 

It is also worth noting here that while both 210 and 168 are broken down into their prime factors in the example above, we could have gotten the same answer using any old factors. We could have factored the numbers as shown below and still arrived at the same GCF of 42. But this method is a little tricky, because we have to be able to recognize that the non-common factors (in this case 5 and 4) don’t share any common factors other than 1.

 

 

 

 

Finding the greatest common factor in a set of monomials is not very different from finding the GCF of two whole numbers. Our method remains the same: we will factor each monomial independently, look for common factors, and then multiply them to get the GCF.

 

Let’s begin by finding the greatest common factor of 15b² and 20b. This problem certainly looks a bit different than ones we are used to; instead of having just whole numbers, this one has the variable terms b² and b present as well. Let’s factor these monomials first, remembering that the variable terms should be factored as well. Can you spot any common factors between 15b² and 20b?

 

 

The monomials have the factors 5 and b in common, which means their greatest common factor is 5 • b, or simply 5b.

 

Let’s try one more problem before moving on: finding the greatest common factor of 81c⁴d and 45c²d. We will first factor 81c⁴d and 45c²d, then look for common factors. Notice that these monomials each have two variables.

 

 

This example presented 81c⁴d and 45c²d as products of prime factors. As you become more familiar with factoring monomials, you will begin to see common factors without needing to reduce monomials to their prime factors. For example, you may have factored the monomials like this:

 

 

Notice that this factoring still leads to the same GCF: 9c²d.

 

Find the greatest common factor of 56xy and 16y3.   A) 8   B) 8y   C) 16y   D) 8xy3   Show/Hide Answer A) Incorrect. 8 is a common factor, but you need to account for the variable terms that the two monomials have in common as well. The correct answer is 8y.   B) Correct. The expression 56xy can be factored as 8 • 7 • x • y, and 16y3 can be factored as 8 • 2 • y • y • y. They have the factors 8 and y in common. Multiplying them together will give you the GCF: 8y.   C) Incorrect. y is a common factor, but 16 is not; 56 is not evenly divisible by 16! Think of numbers that are factors of both 56 and 16. The correct answer is 8y.   D) Incorrect. 8 is a common factor, but you need to account for the variable terms that the two monomials have in common as well. The entire term xy3 is not a factor of either monomial. The correct answer is 8y.

 

 

 

When two or more polynomials are combined (either added or subtracted), the resulting expression is called a polynomial. We can factor polynomials using some of the same logic that we applied to factoring monomials.

 

Remember that factoring is the process of breaking a number down into its multiplicative components. As you look at the examples of simple polynomials below, try to identify factors that the terms of the polynomial have in common.

 

 

To factor a polynomial, first identify the greatest common factor of the monomial terms. Then use the Distributive Property to rewrite the polynomial as the product of the GCF and the other parts of the polynomial. (Remember, the distributive property says that a(b + c) = a • b + a • c.) This rewritten expression is the factored form of a polynomial.

 

 

Look at how we can “pull out” the common factor in each polynomial. We know that both terms in a polynomial are divisible by their GCF, so we can rewrite each polynomial as a product of the GCF and the combined "left over" factors of each monomial.

 

Let's go through the process of factoring a polynomial, step by step.

 

 

To reassure ourselves that this is correct, we can multiply 6d • (4d – 3), checking to see if we get the original form of the polynomial, 24d² – 18d.

 

The factored form of a polynomial is one in which the polynomial is written as a product of factors, and each non-monomial factor has no common factors in its terms. For example, the factors of 6d(4d – 3) are 6, d, and 4d – 3. The two terms of 4d – 3 have no common factors.

 

As a final example, let’s return to a set of monomials for which we found the GCF earlier on: 81c⁴d and 45c²d. If we add these two quantities and make a single polynomial, how could we factor it?

 

 

Knowing the greatest common factor of the two monomials helped us factor the polynomial as a whole. Having identified the GCF, we use the Distributive Property to arrive at the factored form: 9c²d(9c² + 5).

 

Factor: 8r4 – 11r3.   A) 88(r4 – r3)   B) 8r(r3 – 3)   C) r3(r – 1)   D) r3(8r – 11)   Show/Hide Answer A) Incorrect. 88 is the lowest common multiple, not the greatest common factor, of 11 and 8. If 88(r4 – r3) were expanded it would become 88r4 – 88r3, not 8r4 – 11r3. The correct answer is r3(8r – 11).   B) Incorrect. 8 is not a common factor of 8 and 11. If 8r(r3 – 3) were expanded it would become 8r4 – 24r, not 8r4 – 11r3. The correct answer is r3(8r – 11).   C) Incorrect. r3 is a common factor, but the values 8 and 11 have been left out of this factorization. If r3(r – 1) were expanded it would become r4 – r3, not 8r4 – 11r3. The correct answer is r3(8r – 11).   D) Correct. The values 8 and 11 share no common factors, but the GCF of r4 and r3 is r3. Using the Distributive Property, we can rewrite the polynomial as r3(8r – 11).

 

 

 

Any monomial or polynomial can be expressed as a product of factors. We use some of the same technique that we apply to factoring integers to factor polynomials. To factor a polynomial, identify the greatest common factor of the monomial terms, and then use the Distributive Property to rewrite the expression. Once a polynomial in a • b + a • c form has been rewritten as a(b + c), where a is the GCF, we say that the polynomial is in factored form.