In this video, we will learn how to
write algebraic expressions as a product of irreducible factors.
We’ll begin with the understanding
that factoring an algebraic expression is the reverse process of expanding a product
of algebraic factors. So we can use the process of
expanding in reverse to factor many algebraic expressions. For example, we can expand 𝑥 times
𝑥 plus two by distributing the factor of 𝑥 as follows. First, we take the product of 𝑥
and 𝑥 to get the first term. Then, we take the product of 𝑥 and
two to get the second term.
By simplifying this expression, we
obtain 𝑥 squared plus two 𝑥. If we write this process in
reverse, we have 𝑥 squared plus two 𝑥 equals 𝑥 times 𝑥 plus 𝑥 times two equals
𝑥 times 𝑥 plus two. Starting with 𝑥 squared plus two
𝑥, we note that both terms share a factor of 𝑥. Then, we take this shared factor
out from each term to get 𝑥 times 𝑥 plus two. These are the irreducible factors
of 𝑥 squared plus two 𝑥. We can follow this same process to
factor any algebraic expression in which every term shares a common factor.
Let’s clear some space to factor a
multivariable expression, four 𝑥𝑦 plus two 𝑥 squared 𝑦 minus six 𝑥. We want to check for common factors
of all three terms, which we can start doing by checking for a common constant
factor. We see that four, two, and negative
six all share a common factor of two. In fact, this is the greatest
common factor of the three coefficients.
Next, we want to also check for
common factors of variables. We are looking for nonzero powers
of 𝑥 and 𝑦 in each term. We note that all three terms
contain a nonzero power of 𝑥. But only the first two terms
contain a nonzero power of 𝑦. We cannot take out more than the
lowest power as a factor. So the greatest shared factor of a
power of 𝑥 is just 𝑥. There is no nonzero power of 𝑦
that is common to all three terms.
Altogether, we find that the common
factors are two and 𝑥, which we multiply to give two 𝑥. This is the greatest common factor
of the three terms. We know that we’ve found the GCF
when we cannot take out any further common factors.
We want to take the factor of two
𝑥 out of the expression. This is done separately for each
term. Four 𝑥𝑦 factors as two 𝑥 times
two 𝑦. Two 𝑥 squared 𝑦 factors as two 𝑥
times 𝑥𝑦. And negative six 𝑥 factors as two
𝑥 times negative three. What remains of each term after
dividing by two 𝑥 is placed in parentheses. Thus, we factor the original
expression to get two 𝑥 times two 𝑦 plus 𝑥𝑦 minus three. Since factoring is the reverse of
expanding a product, we may check our factorization by distributing the GCF into the
parentheses. In our next example, we’ll follow
this process to factor an algebraic expression by identifying the greatest common
factor of its terms.
Factorize 12𝑥 cubed 𝑦 squared
minus eight 𝑥𝑦 cubed fully.
To factor this expression, we need
to find the greatest common factor of the constant coefficients and each variable
separately. Let’s start with the
coefficients. In this case, we want to find the
greatest common factor of 12 and negative eight. We can see that 12 equals four
times three and negative eight equals four times negative two. And since three and negative two
share no common factors other than one, we can say that four is the greatest common
factor of the coefficients.
Let’s now check each term for
factors of powers of 𝑥 and 𝑦. We know that we cannot take out
more than the lowest power of each variable in common to both terms. We see that both terms have a power
of 𝑥. Since the lowest power of 𝑥 is
one, we know that the greatest common factor of a power of 𝑥 is just 𝑥. As for 𝑦, we see that both terms
have a power of 𝑦 and that the lowest power of 𝑦 is 𝑦 squared. Therefore, the greatest shared
factor of a power of 𝑦 is 𝑦 squared.
Finally, we can multiply the four,
𝑥, and 𝑦 squared together to find the greatest common factor of the terms. Now we’ll factor four 𝑥𝑦 squared
from each term separately. For the first term, we have four
𝑥𝑦 squared times three 𝑥 squared. For the second term, we have four
𝑥𝑦 squared times negative two 𝑦. This allows us to take out the
factor of four 𝑥𝑦 squared as follows, which means the full factorization of 12𝑥
cubed 𝑦 squared minus eight 𝑥𝑦 cubed is four 𝑥𝑦 squared times three 𝑥 squared
minus two 𝑦.
Next, we’ll examine the process of
expanding the product of two linear factors to help us understand the reverse
process of factoring a quadratic expression.
Let’s begin with an example of
expanding two linear factors. Then, we’ll reverse the
process. For example, if we expand the
product of two 𝑥 plus three and 𝑥 plus two, we get 𝑥 times two 𝑥 plus three plus
two times two 𝑥 plus three equals two 𝑥 squared plus three 𝑥 plus four 𝑥 plus
six, which simplifies to two 𝑥 squared plus seven 𝑥 plus six.
To reverse this process, we would
start with the quadratic expression and work backwards to write it as two linear
factors. However, we might wonder how we’d
know to split the coefficients of 𝑥 into negative three 𝑥 and four 𝑥. We can do this by noticing special
qualities of three and four. Not only do they add to equal the
coefficient of 𝑥, they also multiply to equal the product of the leading
coefficient and the constant. That is, we see that the product of
two and six equals 12, just as three times four equals 12.
To put this in general terms, for a
quadratic expression in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, we have identified
a pair of real numbers 𝑚 and 𝑛 such that the product of 𝑎 and 𝑐 equals 𝑚 times
𝑛 and 𝑏 equals 𝑚 plus 𝑛. In our example, 𝑚 equals three and
𝑛 equals four. So we can write two 𝑥 squared plus
seven 𝑥 plus six as two 𝑥 squared plus three 𝑥 plus four 𝑥 plus six.
Next, we’ll factor the first two
terms and the last two terms separately. Taking out the GCF of 𝑥 from the
first two terms gives us 𝑥 times two 𝑥 plus three. Taking out a GCF of two from the
second pair of terms gives two times two 𝑥 plus three. So we have the equivalent
expression 𝑥 times two 𝑥 plus three plus two times two 𝑥 plus three.
Now, we see the common factor of
two 𝑥 plus three, which we take out from both terms, resulting in the two linear
factors of the quadratic expression, two 𝑥 plus three and 𝑥 plus two. With this property in mind, let’s
examine a general method that will allow us to factor any quadratic expression that
We can factor a quadratic
polynomial of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 using the following steps. First of all, we calculate the
product of 𝑎 and 𝑐, then list its factor pairs. We want to find all the pairs of
numbers 𝑚 and 𝑛 such that 𝑚 times 𝑛 equals the product of 𝑎 and 𝑐. We note that if the product 𝑎𝑐 is
positive, its factors 𝑚 and 𝑛 are either both positive or both negative. However, if 𝑎𝑐 is negative, 𝑚
and 𝑛 must have alternating signs.
In the second step, we’ll need to
identify the pair of factors of 𝑎𝑐 that also add to give 𝑏, the coefficient of
𝑥. That is, 𝑚 plus 𝑛 equals 𝑏. Then, we are ready to rewrite the
𝑥-term using these factors. This means the 𝑥-term is split
into 𝑚𝑥 and 𝑛𝑥.
In the next step, we will factor
the first two terms and the final two terms separately. After step four, we should have a
common factor in the parentheses. The last step is to take out the
common factor. This process is much simpler when
𝑎 equals one, as we will see in our next example.
Factor 𝑥 squared plus eight 𝑥
We’re asked to factor a quadratic
expression with a leading coefficient one. We can do this by finding two
numbers 𝑚 and 𝑛 whose sum is the coefficient of 𝑥, eight, and whose product is
the constant, 12. One way of finding 𝑚 and 𝑛 is to
list the factor pairs of 12 like this. Since the product of 𝑚 and 𝑛 is
positive and the sum is positive, the factors must both be positive. We see that two plus six equals
eight. So 𝑚 equals two and 𝑛 equals
We use 𝑚 and 𝑛 to rewrite the
𝑥-term in the quadratic expression. So we have 𝑥 squared plus two 𝑥
plus six 𝑥 plus 12. We note that the first two terms
share a factor of 𝑥 and the last two terms share a factor of six. Taking out these factors yields 𝑥
times 𝑥 plus two plus six times 𝑥 plus two. We note that both terms share a
factor of 𝑥 plus two.
Just as we took out the greatest
common factor of 𝑥 or six, here we take out the greatest common factor of 𝑥 plus
two from both terms, which gives the linear factors 𝑥 plus two and 𝑥 plus six. We have shown that the
factorization of the quadratic expression 𝑥 squared plus eight 𝑥 plus 12 is 𝑥
plus two times 𝑥 plus six. For comparison, we see that this
process returns the same factors, even if we swap the order of the 𝑥-terms. Because multiplication is
commutative, the factors can be written as 𝑥 plus two times 𝑥 plus six or 𝑥 plus
six times 𝑥 plus two.
In our next example, we’ll apply
this method to fully factor a nonmonic cubic expression. “Nonmonic” means the leading
coefficient is not one.
Factorize fully six 𝑥 cubed minus
17𝑥 squared plus 12𝑥.
We note that this expression is a
cubic, since the highest power of 𝑥 is 𝑥 cubed. When factoring cubics, we should
first try to identify whether there is a common factor of 𝑥 we can take out. Although there is no coefficient
factor shared by all three terms, we see that the lowest nonzero shared power of 𝑥
is 𝑥. Thus, the greatest common factor is
just 𝑥. We can take out this factor to
yield 𝑥 times six 𝑥 squared minus 17𝑥 plus 12.
Now, we recall that to factor a
quadratic of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, we’ll need to find two real
numbers 𝑚 and 𝑛 whose product is 𝑎𝑐 and whose sum is 𝑏. In our case, we have 𝑎 equals six,
𝑏 equals negative 17, and 𝑐 equals 12. So we want two numbers that sum to
give negative 17 and multiply to give six times 12, which is 72.
Since the product is positive and
the sum is negative, 𝑚 and 𝑛 must both be negative. So we will check all the negative
factor pairs of 72. We find the only pair that adds to
negative 17 is negative eight and negative nine. So we use these two numbers to
rewrite the 𝑥-term, negative 17𝑥, as negative eight 𝑥 minus nine 𝑥. Actually, the order in which we
write these does not matter, as the method still works, even if we decide to write
negative nine 𝑥 minus eight 𝑥. Then, we can factor the first two
terms and the last two terms separately. Specifically, we can take out the
shared factor of two 𝑥 from the first pair and three from the second pair. This looks like 𝑥 times two 𝑥
times three 𝑥 minus four plus three times negative three 𝑥 plus four.
To make the two terms share a
factor, we need to take a factor of negative one out of the second term to obtain 𝑥
times two 𝑥 times three 𝑥 minus four minus three times three 𝑥 minus four. Finally, we take out the shared
factor of three 𝑥 minus four. So, including the first common
factor of 𝑥, the full factorization of six 𝑥 cubed minus 17𝑥 squared plus 12𝑥 is
𝑥 times three 𝑥 minus four times two 𝑥 minus three.
There are many other methods we can
use to factor quadratics. Let’s looks at another example
where we’ll apply this process using substitution.
Factorize fully 𝑦 to the fourth
power minus five 𝑦 squared minus 14.
We want to fully factor the given
expression. However, we can see that this is
not a quadratic, because the highest power of 𝑦 is four. Usually, we would search for a
factor shared by all three terms, but there is none. However, we can treat this
expression as a quadratic if we substitute 𝑥 for 𝑦 squared.
First, we note that 𝑦 to the
fourth power is 𝑦 squared to the power of two. Then, we can temporarily rewrite
the expression as 𝑥 squared minus five 𝑥 minus 14, as shown. We can now factor this expression
by identifying two numbers whose product is the constant, negative 14, and whose sum
is the coefficient of the 𝑥-term, negative five. We can find these by considering
the factors of negative 14. We see that two plus negative seven
equals negative five. So we’ll use those values to split
the 𝑥-term. That looks like 𝑥 squared minus
seven 𝑥 plus two 𝑥 minus 14. Or we may write two 𝑥 before
negative seven 𝑥.
Next, we take the shared factor of
𝑥 in the first two terms and the shared factor of two in the final two terms to
obtain the equivalent expression 𝑥 times 𝑥 minus seven plus two times 𝑥 minus
seven. Then, we can take out the shared
factor of 𝑥 minus seven from the two terms to get 𝑥 minus seven times 𝑥 plus two,
or equivalently 𝑥 plus two times 𝑥 minus seven.
Since the original expression was
given in terms of 𝑦, let’s substitute 𝑦 squared back in for 𝑥 in our factored
expression. Therefore, the full factorization
of 𝑦 to the fourth power minus five 𝑦 squared minus 14 in terms of 𝑦 is 𝑦
squared plus two times 𝑦 squared minus seven.
Before our last example, let’s look
at a special case called the difference of squares. To understand the difference of two
squares’ factoring property, let’s consider the product of two binomials: 𝑥 plus 𝑦
and 𝑥 minus 𝑦. After fully expanding the product
using distribution, we note that the terms 𝑥𝑦 and negative 𝑥𝑦 sum to give zero,
which leads to an expression with only two terms: 𝑥 squared minus 𝑦 squared. This result is called the
difference of two squares.
By applying the above steps in
reverse, we arrive at a way to factor any such expression. Any algebraic expression of the
form 𝑥 squared minus 𝑦 squared can be factored as 𝑥 plus 𝑦 times 𝑥 minus
𝑦. In our next example, we’ll use this
property to factor a given quadratic expression.
Factor the expression 𝑥 squared
We first note that the expression
we’re asked to factor is the difference of two squares, specifically 𝑥 squared
minus seven squared. We recall that any expression of
the form 𝑥 squared minus 𝑦 squared has factors 𝑥 plus 𝑦 and 𝑥 minus 𝑦. We take 𝑦 equal to seven. A common mistake would be to use 𝑦
equals 49, but this would not be correct. We must identify the number whose
square is 49, and that is seven.
Therefore, we have 𝑥 squared minus
49 equal to 𝑥 squared minus seven squared, which equals 𝑥 plus seven times 𝑥
minus seven. This is the full factorization of
the quadratic expression 𝑥 squared minus 49.
Let’s finish by recapping some of
the important points from this video. The greatest common factor of an
algebraic expression is the greatest common factor of the coefficients multiplied by
each variable raised to the lowest exponent in which it appears in any term. We can factor an algebraic
expression by checking for the greatest common factor of all its terms and taking
this factor out. We can factor a quadratic in the
form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 by finding two numbers whose product is 𝑎𝑐 and
whose sum is 𝑏. We use these two numbers to rewrite
the 𝑥-term and then factor the first pair and final pair of terms. Finally, we factor the whole
If we’re asked to factor a cubic or
higher-degree polynomial, we should first check if each term shares any common
factors of the variable to simplify the expression. Or we may be able to use
substitution to rewrite the expression as a quadratic. And finally an expression of the
form 𝑥 squared minus 𝑦 squared is called a difference of two squares. We can factor this as 𝑥 plus 𝑦
times 𝑥 minus 𝑦.