- Welcome to basic factoring. In the last lesson, we multiplied polynomials, this lesson runs that process in reverse. Instead of starting with factors and expanding them into a polynomial, we'll start with a polynomial and break it back into the factors that created it.

Factoring is one of the most important algebra skills on the SAT because it shows up in solving equations, simplifying expressions and recognizing patterns. Let's kick this off with some framing. A helpful way to think about factoring is that it is reverse multiplying.

For example, if you multiply the quantity "x" plus three times the quantity "x" plus five, you get "x" squared plus "8x" plus 15. Factoring asks the reverse question. If I start with "x" squared plus "8x" plus 15, what factors produced it?

That's the mindset we'll use throughout this lesson. Now, let's talk about the first factoring technique, the greatest common factor, otherwise known as GCF. The GCF is the largest factor that divides every term in an expression. When a GCF exists, we wanna pull it out and make the expression inside the parentheses smaller and easier to work with.

Finding the greatest common factor has two parts. For the coefficients, find the largest number that divides all of them. For the variables, find the lowest power that appears in every term. Let's try one. "5x" plus 45.

Before we reveal the answer, take a moment to see if you can identify the GCF. What's the largest number that divides both terms evenly? The answer is five. Both five and 45 are divisible by five. Notice that the second term doesn't contain an "x," so "x" is not part of the GCF.

That means "5x" plus 45 in its factored form is five times the quantity "x" plus nine. A quick distribution check confirms our work, five times "x" is "5x," five times nine is 45, everything checks out. Okay, let's make it slightly more interesting.

"3x" cubed plus "12x." This time, the GCF includes a variable. Before we reveal it, take a moment to see if you can identify both the numerical part and the variable part. Okay, here we go.

The coefficients three and 12 share a GCF of three. For the variables, both terms contain at least one "x." The lowest power shared by both terms is just "x" to the first power, so the greatest common factor is "3x." Now, we want to factor out the GCF to arrive at "3x" times the quantity "x" squared plus four.

Now, let's lock in one of the most important habits in factoring. Whenever you begin a factoring problem, check for a greatest common factor first. Even if the expression appears to fit another factoring pattern, pulling out the greatest common factor first usually makes everything easier.

All right, let's move to the second factoring technique, factoring quadratics of the form "x" squared plus "bx" plus "c." The goal is to rewrite the quadratics as two binomials, "x" plus "p" times "x" plus "q." The question becomes, how do we find "p" and "q?" Well, here's the key relationship.

The two numbers must add to the middle coefficient and multiply to the constant term at the end, that's it. If you can find two numbers that satisfy both condition, you found the factors. That relationship comes directly from the foil multiplication patterns we talked about in the last lesson, but for SAT purposes, the important thing to remember is simply the sum gives you the middle coefficient, the product gives you the constant term.

Let's try one. Factor "x" squared plus "8x" plus 15. Before we work through it, pause for a second, can you think of two numbers that add to eight and multiply to 15? Three and five are the numbers we want. Three plus five is eight, three times five is 15, that means "x" squared plus "8x" plus 15 factors into the quantity "x" plus three times the quantity "x" plus five, and if you're ever unsure, you can always foil it back to verify.

Let's move on and try a mixed-sign example. Factor "x" squared plus "4x" minus 21. First, notice that the product is negative, that means one number must be positive and the other must be negative. Now, let's search for the factor pairs of 21, one and 21, three and seven, and since we need a positive sum of four, that means we need a positive seven and a negative three, their sum is four, their product is a negative 21, so the factored form is the quantity "x" plus seven times the quantity "x" minus three.

Here is one more. Factor "x" squared minus "16x" plus 48. This time, the product is positive, the sum is negative. That tells us both numbers must be negative. The pair we need is negative four and negative 12, so the factorization becomes the quantity "x" minus four times the quantity "x" minus 12.

Now, let's look at our third factoring technique, difference of squares. The key is recognizing the form "a" squared minus "b" squared, no middle term, just one perfect square minus another. The factoring formula for difference of squares is "a" squared minus "b" squared equals the quantity "a" plus "b" times the quantity "a" minus "b." Same two terms, opposite signs. That's the pattern.

Here is an example. "x" squared minus 49. Notice that 49 is seven squared, so we have "x" squared minus seven squared, there's your difference of squares. So the factorization is the quantity "x" plus seven times the quantity "x" minus seven.

Now, here's one with a coefficient. "9x" squared minus 16. Both terms are still perfect squares. "9x" squared equals "3x" squared, 16 equals four squared. If we apply the difference of squares factor formula, we have the quantity of "3x" plus four times the quantity of "3x" minus four and we're done.

All right, let's put everything together into a factoring playbook. Step one, check for a greatest common factor, step two, identify the pattern, step three, apply the appropriate factoring technique, step four, verify if needed.

This sequence will solve a huge percentage of SAT factoring problems. Now, it's your turn. Pause the lesson and try the five practice problems on the screen, take your time and resume when ready.

All right, let's check the answers. If you got all five, very well done, if you missed any, revisit the part of the lesson that covers that topic. Let's wrap with a summary of basic factoring. Factoring is reverse multiplying.

Always check for a GCF or greatest common factor first. The GCF habit alone prevents a lot of factoring mistakes and can save a lot of time. For "x" squared plus "bx" plus "c," find two numbers that add to "b" and multiply to "c." A lot of factoring questions come down to that relationship, and recognizing we have a difference of squares pattern can save a lot of time.

That's gonna do it for basic factoring, up next, we'll add a few more factoring tools to the toolbox, including cases where the coefficient of "x" squared isn't one and another important factoring pattern.