Factoring - GCF
So now we can talk about some factoring techniques. In the previous two lessons we made extensive use of the distributive law. The distributive law is perhaps the most important basic arithmetic process that we can talk about here. So, here's the statement of the distributive law, multiplication distributing over addition.
And notice, that when we start on the left side and move to the right, we say we're distributing, we're distributing P. But of course this is an equation, we can use it either way. When we start on the right side and move to the left. So we express P now as a factor of the whole thing rather than a factor of each individual term, that is called factoring out.
It's very important to recognize the distributing and factoring out are two sides of the same coin. It's the same fundamental process. We just have two different words for depending on which way we're going in that process. So that's very important to understand.
In algebra, factoring is a big topic. Factoring means rewriting any expression as a product of two or more factors. As we will see in the section on algebraic equations, factoring is one powerful equation solving strategy. So this is a very important skill that well use in solving equations. Before we can employ this strategy in solving equations, we need to understand the various kinds of factoring techniques, so we have a few videos devoted to factoring techniques.
This is the easiest of the factoring techniques, simply factoring out a greatest common factor from a binomial. Now if you haven't seen the idea of a greatest common factor before, I highly recommend going back to the integer properties video where we talked about the idea of greatest common factor with just ordinary integers. This expands the idea to algebra.
So for example, if we have the binomial 5x plus 45, just a simple linear binomial. Clearly, those two numbers have a greatest common factor of 5. So we could factor out a 5, and write this as 5(x + 9). Here's another binomial. This one is a cubic binomial. Here we can factor out both a 3 from the coefficients.
3 is the greatest common factor of the coefficients. And also, because we have three factors of x in the first term, and one factor of x in the second term, we can factor out an x. So we factor out 3x. And that leaves us in the first term with 3x squared and on the second term, it leaves us just with 4.
So that is a totally factored expression This expression, very tricky. Probably the powers here are a little bit higher than you're gonna see except on very, very advanced problems on the test. But theoretically the same principles here, the highest power of x is x to the 5th so we can factor out an x to the 5th on both terms.
The highest power of y is y to the 4th so we can factor out a y to the 4th in both terms. And that leaves us with x to the 5th times x to the 4th times the quantity x squared plus y squared. X squared plus y squared those are what's left of the two factors once we factor out x to the 5th and y to the 4th.
We could also factor out a greatest common factor from a trinomial. So here's a trinomial, this turns out as a trinomial where the highest power is x to the 4th. And notice that among the coefficients, the greatest common factor is 2, so we can factor out a 2. And the lowest power of x is x squared, so we can factor out an x squared from all the other Xs.
And so we factor out 2x squared and that leaves us with just x squared in the first term, just an x in the second term, and simply a 5 in the third term. So we factor this down in to 2x squared times a quadratic trinomial. Here's some practice problems. Pause the video here, and factor out the greatest common factor. And these are the answers.
In this video we discussed the factoring technique of factoring out a greatest common factor.
Read full transcriptAnd notice, that when we start on the left side and move to the right, we say we're distributing, we're distributing P. But of course this is an equation, we can use it either way. When we start on the right side and move to the left. So we express P now as a factor of the whole thing rather than a factor of each individual term, that is called factoring out.
It's very important to recognize the distributing and factoring out are two sides of the same coin. It's the same fundamental process. We just have two different words for depending on which way we're going in that process. So that's very important to understand.
In algebra, factoring is a big topic. Factoring means rewriting any expression as a product of two or more factors. As we will see in the section on algebraic equations, factoring is one powerful equation solving strategy. So this is a very important skill that well use in solving equations. Before we can employ this strategy in solving equations, we need to understand the various kinds of factoring techniques, so we have a few videos devoted to factoring techniques.
This is the easiest of the factoring techniques, simply factoring out a greatest common factor from a binomial. Now if you haven't seen the idea of a greatest common factor before, I highly recommend going back to the integer properties video where we talked about the idea of greatest common factor with just ordinary integers. This expands the idea to algebra.
So for example, if we have the binomial 5x plus 45, just a simple linear binomial. Clearly, those two numbers have a greatest common factor of 5. So we could factor out a 5, and write this as 5(x + 9). Here's another binomial. This one is a cubic binomial. Here we can factor out both a 3 from the coefficients.
3 is the greatest common factor of the coefficients. And also, because we have three factors of x in the first term, and one factor of x in the second term, we can factor out an x. So we factor out 3x. And that leaves us in the first term with 3x squared and on the second term, it leaves us just with 4.
So that is a totally factored expression This expression, very tricky. Probably the powers here are a little bit higher than you're gonna see except on very, very advanced problems on the test. But theoretically the same principles here, the highest power of x is x to the 5th so we can factor out an x to the 5th on both terms.
The highest power of y is y to the 4th so we can factor out a y to the 4th in both terms. And that leaves us with x to the 5th times x to the 4th times the quantity x squared plus y squared. X squared plus y squared those are what's left of the two factors once we factor out x to the 5th and y to the 4th.
We could also factor out a greatest common factor from a trinomial. So here's a trinomial, this turns out as a trinomial where the highest power is x to the 4th. And notice that among the coefficients, the greatest common factor is 2, so we can factor out a 2. And the lowest power of x is x squared, so we can factor out an x squared from all the other Xs.
And so we factor out 2x squared and that leaves us with just x squared in the first term, just an x in the second term, and simply a 5 in the third term. So we factor this down in to 2x squared times a quadratic trinomial. Here's some practice problems. Pause the video here, and factor out the greatest common factor. And these are the answers.
In this video we discussed the factoring technique of factoring out a greatest common factor.
