单项式因式分解

Examples: Factoring Monomials

Factoring monomials is a process of breaking down a monomial into smaller terms. Although factoring a polynomial usually shortens the polynomial, factoring a monomial expands it.

Let's start by reviewing the definition of a monomial.

Monomial: An algebraic expression consisting of one term

For example, the expression $2x$ is a monomial, but the expression $x+2$ is a polynomial.

Let's identify the parts a monomial might have:

Some monomials might have more than one variable, for example:

Factoring a monomial

When you factor a monomial, you are breaking it down into smaller terms. Think of it as expressing the monomial as a product of smaller monomials!

For example, say we want to factor $x^2y^3.$ How can we express this monomial in terms of smaller monomials? There are actually many ways to do this! Find factors of the different components of the monomial (like the factors of $x^2$ and $y^3$) and split up the monomial accordingly. Here are some examples: $$(x^2y)(y^2)$$ $$(x)(xy^3)$$ $$(x^2y^2)(y)$$

Notice how if you multiply each factor together, you get the original monomial, $x^2y^3.$

Completely Factoring a Monomial

To completely factor a monomial, we expand each part of the monomial until we cannot expand anymore. For example:

What if we have a coefficient in a monomial?

We write the coefficient in its prime factorization form. For example:

How Do You Factor a Monomial?

First, let's remind ourselves of what a monomial is: $$\text{Monomial examples}$$ $$3x$$ $$x$$ $$-2x$$ $$4x^2$$

Monomials have only one term. When we're asked to factor a monomial, we're really being asked to EXPAND the monomial.

Think of a monomial ($20b^3c^2$, for example) as an accordion, all folded up. When we factor a monomial, we are unfolding that accordion, stretching it out.

Factorization

When you factor a monomial, you are breaking it down into smaller terms. It's like expressing your monomial in terms of smaller monomials!

For example, say we want to factor $20b^3c^2.$ How can we express this monomial using smaller terms? Well, there are multiple ways! Just think about the factors of the different parts of the monomial (like the factors of $20,b^3,$ and $c^2$) and split up the monomial accordingly. Here are a few examples: $$(5)(4)b^3c^2$$ $$(20b^2)(bc^2)$$ $$(10)(2b^3c^2)$$

Notice how if you multiply each factor together, you get the original monomial, $20b^3c^2.$

Complete Factorization

You might be asked to find a monomial's prime factorization. This is the same thing as being asked to completely factor it.

So, let's quickly go over what prime numbers are, in order to factor monomials. A prime number is a natural number greater than $1$ that is only divisible by 1 and itself. Nothing else divides into it cleanly.

Examples of prime numbers include $2, 3, 5, 7, 11, 13$ and many more. Notice that each of these numbers only has two natural-number factors: $1$ and itself.

Let's work through an example problem step-by-step to show how we go about factoring monomials:

$$\text{Completely factor the following monomial: } 20b^3c^2$$

STEP 1: Break the monomial's coefficient down into all of its prime factors

First, we need to break our monomial's coefficient down into all of its prime factors. In order to do this, we're going to create a factor tree, a diagram that will help us see exactly which prime numbers we need to multiply together to produce that coefficient.

We start building our factor tree by writing down our original coefficient, $20$. We then draw two lines leading away from the coefficient, one going down and to the left, and one going down and to the right. You can think of these two lines as "branches" in our factor tree.

That's right, it's $2$!

This means that $2$ will be the first prime number in our factor tree. Let's write down this number under the left branch that we just drew.

Next, we take the number at the junction of our two branches, divide it by the prime number that we just wrote down, and write down that result under the right branch. In this case, we take $20$, divide it by $2$, and write down our result (i.e. $10$) under the right branch.

At this point, our factor tree should look a little something like this:

In order to continue our tree, we shift our attention to the number that we wrote down under our right branch: $10$.

Right again! We can now write down the number $2$ under our $10$ branch. Then, since $10 / 2 = 5$, we write the number $5$ under our $10$ branch.

Now our factorization tree looks like this:

STEP 2: Break down the monomial's variables and exponents

Similarly, we want to expand the variables in our original monomial. Since a variable's exponent is essentially shorthand for how many times that variable is being multiplied by itself, when we "expand" a variable, all we have to do is write out all of that abbreviated multiplication:

$$b^3=b\cdot b \cdot b$$ $$c^2= c \cdot c$$

So, when we fully factor and expand the monomial $20b^3c^2$, we get the following:

$$20b^3c^2 = 2\cdot 2 \cdot 5 \cdot b\cdot b\cdot b\cdot c\cdot c$$

Let's do another example!

Solution

$$\begin{align} 16r^4t &=(2\cdot 2\cdot 2\cdot 2)r^4t\\ &= (2\cdot 2\cdot 2\cdot 2)(r\cdot r\cdot r\cdot r)t\\ &= 2\cdot 2\cdot 2\cdot 2 \cdot r\cdot r\cdot r\cdot r \cdot t \end{align}$$

(Video) Factor Monomials

by Chris Raymaakers and Red River College Wise Guys

The above video by Chris Raymaakers describes how to expand a monomial.

$\large{\text{Summary}}$

Completely factoring, or completely expanding, a monomial is simple. First, we write out the prime factorization of the coefficient. Then, we write all the variables out, instead of using exponents.

In the video, they look at how to factor $-18g^4h$.

Step 1 is to take prime factorization of the coefficient, $-18$. We can rewrite this as $-1\times18$. $$-18=-1\times18$$ Then, we can break $18$ into $2$ and $9$. $$\begin{array}{ccccc} &&18&&\\ &\diagup&&\diagdown&\\ \boxed{2}&&&&9 \end{array}$$ Since $2$ is a prime, we can stop that branch. But $9$ can be broken up into $3$ and $3$. $$\begin{array}{ccccccc} &&18&&&&\\ &\diagup&&\diagdown&&&\\ \boxed{2}&&&&9&&\\ &&&\diagup&&\diagdown&\\ &&\boxed{3}&&&&\boxed{3} \end{array}$$ So the prime factorization of $-18$ is: $$-18=-1\times2\times3\times3$$

Step 2 is to write out all the variables without using exponents. $$g^4h=g\cdot g\cdot g\cdot g\cdot h$$

Finally, we can write the whole thing out. The factor of $-18g^4h$ is: $$-18g^4h=-1\cdot2\cdot3\cdot3\cdot g\cdot g\cdot g\cdot g\cdot h$$

Note

Factoring a monomial is not the same as completely factoring it. Completely factoring a monomial is writing down its entire prime factorization. Factoring a monomial is more general: it just means breaking down the monomial into smaller terms...these smaller terms do not have to be completely factored themselves. For example:

Monomial: $-18g^4h$

Completely factored monomial: $-1\cdot2\cdot3\cdot3\cdot g\cdot g\cdot g\cdot g\cdot h$

Factored monomial: $-2\cdot 9\cdot g^4 \cdot h = (-2)(9)g^4h$

Notice that each part of the completely factored monomial, $-1\cdot2\cdot3\cdot3\cdot g\cdot g\cdot g\cdot g\cdot h,$ is completely factored. For example, you can't factor $2$ or $g$ any more than it already is.

Also notice that each part of the factored monomial, $(-2)(9)g^4h,$ is not necessarily completely factored. For example, $h$ is completely factored, but $9$ is not.