Factoring Polynomials

What Is Factoring Polynomials?

Factoring a polynomial is the process of rewriting it as a product of two or more simpler polynomials, called its factors. In essence, you are 'un-distributing' to find which expressions were multiplied together to create the original polynomial. This is very similar to finding the prime factors of a number. For example, the number $12$ can be factored into $2\times 6$, but its prime factorization is $2\times 2\times 3$. These are the fundamental building blocks of $12$.

Similarly, when we factor a polynomial like $2x^2+4x$, we look for the simplest 'building blocks' that multiply to produce it. In this case, we can see that both terms share a factor of $2x$. By dividing each term by $2x$, we find the factorization is $2x(x+2)$. The expressions $2x$ and $(x+2)$ are the factors. Just as with numbers, the goal is to factor completely, meaning none of the remaining factors can be broken down any further.

Understanding factoring is crucial because it allows us to solve polynomial equations. If we can set a factored polynomial equal to zero, like $(x-3)(x+2)=0$, the Zero Product Property tells us that either $x-3=0$ or $x+2=0$, making the solutions easy to find ($x=3$ and $x=-2$).

The First Rule: Factoring out the Greatest Common Factor (GCF)

The absolute first step in any factoring problem is to check for and factor out the Greatest Common Factor (GCF). The GCF is the largest monomial that divides every single term of the polynomial. Ignoring this step can make the problem much harder or lead to an incomplete answer.

To find the GCF of a polynomial, follow these two steps:

1. Find the GCF of the coefficients: Look at all the numerical coefficients in the polynomial. Find the largest number that divides evenly into all of them.
2. Find the GCF of the variables: For each variable present in every term, take the one with the lowest exponent. If a variable isn't in all the terms, it's not part of the GCF.

Once you've identified the GCF, you 'pull it out' by writing it in front of a set of parentheses. Inside the parentheses, you write what's left of each original term after dividing it by the GCF.

How Do You Factor a Polynomial by Grouping?

When you encounter a polynomial with four terms, the factoring by grouping method is your best bet. This technique works by splitting the polynomial into two pairs of terms, finding the GCF of each pair, and then factoring out a common binomial factor. It might sound complicated, but it's a very structured process.

Here are the steps for factoring by grouping:

1. Group the terms: Arrange the polynomial into two pairs of terms. Usually, grouping the first two and the last two works well. Place parentheses around each pair.
2. Factor out the GCF from each group: Look at the first pair and factor out its GCF. Then, do the same for the second pair. Be very careful with signs when factoring out a negative GCF.
3. Identify the common binomial: After step 2, you should see that the expressions left in the parentheses are identical. This is your common binomial factor.
4. Factor out the common binomial: Treat the common binomial as a single GCF. Factor it out and write the terms that were multiplying it (the GCFs from each group) in a second set of parentheses.

If the binomials in step 3 do not match, try rearranging the original four terms and starting again. If it still doesn't work, the polynomial may be prime (not factorable by this method).

Factoring Trinomials of the Form $x^2+bx+c$

Trinomials are polynomials with three terms, and those in the form $x^2+bx+c$ (where the leading coefficient is $1$) are the most common type you'll encounter. Factoring these involves a bit of a puzzle. You are looking for two numbers that satisfy two conditions simultaneously.

The goal is to find two integers that:

• Multiply to the constant term, $c$.
• Add to the coefficient of the middle term, $b$.

Once you find these two numbers, let's call them $p$ and $q$, the factored form of the trinomial will be $(x+p)(x+q)$. Creating a small table can help organize your search for the right pair of numbers.

Let's look at the signs. If $c$ is positive, both numbers $p$ and $q$ will have the same sign as $b$. If $c$ is negative, the numbers will have opposite signs.

Factoring 'Complex' Trinomials: The AC Method for $a{x}^2+bx+c$

When a trinomial has a leading coefficient (the $a$ in $a{x}^2+bx+c$) other than $1$, the previous 'puzzle' method gets more complicated. The most reliable technique for these is the AC Method, which cleverly combines the product-sum puzzle with factoring by grouping.

Here are the steps for the AC Method:

1. Multiply $a$ and $c$: Calculate the product $a\times c$.
2. Find two numbers: Just like before, find two numbers that multiply to your new value ($ac$) and add to the middle coefficient, $b$.
3. Rewrite the middle term: Split the middle term, $bx$, into two terms using the two numbers you just found. Your trinomial will now have four terms.
4. Factor by grouping: Use the factoring by grouping method on your new four-term polynomial to get the final answer.

This method always works as long as the trinomial is factorable. It's a powerful and systematic approach.

Shortcuts for Special Factoring Cases

Some polynomials have special patterns that allow you to factor them instantly with a formula, saving you time and effort. Recognizing these patterns is a valuable skill.

Difference of Two Squares

This pattern applies to a binomial where two perfect squares are being subtracted. A perfect square is any number or expression that can be written as something squared (e.g., $25=5^2$ or $9x^2=(3x{)}^2$).

For example, to factor $x^2-49$, we see that $a=x$ and $b=7$. The factored form is $(x-7)(x+7)$.

Perfect Square Trinomials

This pattern applies to a trinomial where the first and last terms are perfect squares and the middle term is twice the product of their square roots. There are two versions, depending on the sign of the middle term.

For example, to factor $x^2+10x+25$, we check: Is the first term a perfect square? Yes, $x^2=(x{)}^2$. Is the last term a perfect square? Yes, $25=(5{)}^2$. Is the middle term $2ab$? Let's see: $2(x)(5)=10x$. Yes, it matches! Since the middle term is positive, the factored form is $(x+5{)}^2$.

Common Factoring Mistakes to Avoid

Factoring has several places where small mistakes can happen. Being aware of these common pitfalls can help you double-check your work and improve your accuracy.

• Forgetting the GCF: This is the most common mistake. Always, always check for a GCF first. Factoring $2x^2+20x+50$ is much harder than factoring $2(x^2+10x+25)$.
• Sign Errors: Be extremely careful with positive and negative signs, especially when factoring by grouping or using the AC method. A single misplaced negative sign will change the entire answer.
• Stopping Too Soon: After you factor a polynomial, look at your factors. Can any of them be factored further? For example, factoring $x^4-16$ as $(x^2-4)(x^2+4)$ is correct, but not complete. The $(x^2-4)$ is a difference of squares and can be factored again into $(x-2)(x+2)$. The complete answer is $(x-2)(x+2)(x^2+4)$.
• Mixing up $b$ and $c$: When factoring trinomials, make sure you are looking for numbers that multiply to $c$ (or $ac$) and add to $b$, not the other way around.
• Incorrectly Factoring out a Negative GCF: When you factor out a negative number, remember to flip the sign of every term inside the parentheses. For example, $-5x+10$ becomes $-5(x-2)$, not $-5(x+2)$.

A Summary of Factoring Strategies

When you're given a polynomial to factor, it can be overwhelming to know where to start. Follow this step-by-step strategy to tackle any factoring problem systematically.

1. Factor out the Greatest Common Factor (GCF). This is always the first and most important step. Don't skip it!
2. Count the number of terms in the remaining polynomial.
3. If there are two terms: Check if it fits the Difference of Two Squares pattern ($a^2-b^2$).
4. If there are three terms (a trinomial):
   
   • Is it a Perfect Square Trinomial ($a^{2}2ab+b^2$)? If so, use the shortcut.
   • If not, is the form $x^2+bx+c$? Find two numbers that multiply to $c$ and add to $b$.
   • Is the form $a{x}^2+bx+c$? Use the AC Method.
5. If there are four terms: Try Factoring by Grouping.
6. Check Your Work: After factoring, look at each factor to see if it can be factored again. You are finished only when all factors are prime. You can also multiply your factors back together to see if you get the original polynomial.