Factor Polynomials with a Common Factor
So if we’re given six 𝑥 plus fifteen and we’re asked to factor this, we’re looking for a “greatest common factor” or “GCF.” So we’re looking for a greatest common factor that goes into both six 𝑥 and fifteen. Well we know that they’re both in the three times table; so three goes into them and there isn’t a bigger number. So we’ll write both of these as with a factor of three. So I could say six 𝑥 is three multiplied by two 𝑥 and fifteen is three multiplied by five.
Now using the distributive property, we’re going to take that three out of each term and put it on the outside of the parentheses. And then on the inside, we’ll have what’s left over from each term. We’ll have first of all two 𝑥 and then five. So now we have fully factored the polynomial six 𝑥 plus fifteen by finding a greatest common factor of both terms and putting that outside the parentheses.
Now we must factor the expression seven 𝑥 cubed plus fourteen 𝑥 squared minus thirty-five 𝑥. So again what we’re looking for is a greatest common factor of each of those terms. So looking at the numbers first of all, we can see that each of those numbers are in the seven times table. So the greatest common factor is seven first of all. And then looking at the variables, we can see that each of them have an 𝑥; so the greatest common factor of all of those terms will be seven 𝑥.
Now if we do exactly the same as we did in the previous example and will write each of those terms with a multiplication of seven 𝑥, you can see that the first term will be seven 𝑥 multiplied by 𝑥 and by 𝑥 again — so seven 𝑥 multiplied by 𝑥 squared. The next term, we know that seven multiplied by two is fourteen. So we’ve got seven 𝑥 multiplied by two first of all. And then focusing on the variables, we know that 𝑥 multiplied by 𝑥 is 𝑥 squared. So it’ll be seven 𝑥 multiplied by two 𝑥.
And now our last term, so let’s look at it as a whole term of negative thirty-five 𝑥. And we know that seven multiplied by negative five is negative thirty-five. So now we have seven 𝑥 multiplied by negative five. And now again using the distributive law, we’re going to look at- each of those terms have a seven 𝑥 inside; so we’re going to take seven 𝑥 and put it on the outside of the parentheses. And then on the inside, we’ve got each of those terms in blue. So first of all, we have 𝑥 squared plus two 𝑥 minus five. And now we have it; we have fully factored this polynomial.
So just as a recap of what we did. We looked at each of the terms and we tried to find the greatest common factor. So we looked at the numbers first and we found that seven was common in each of the terms. And then looking for the variables, we could see that 𝑥 was common in each of the terms. So the greatest common factor was seven 𝑥. Then we looked at each of the terms individually and said seven 𝑥 multiplied by what is that previous term. And then once we’ve done that, we used the distributive law to take seven 𝑥 on the outside of the parentheses, leaving us with the terms left over.
Now let’s look at one with not just 𝑥, but also 𝑦. So we have to factor twelve 𝑥 squared 𝑦 to the power of five minus thirty 𝑥 to the power of four 𝑦 squared. So we’re looking first of all for the greatest common factor in numbers. So we can see that two goes into both; well, that’s quite small. So maybe there’s a larger one, but we know three goes in. Then if two and three goes in, then that means that six must go in. And there isn’t a bigger number than that. So we’ve got the greatest common factor as six.
Now looking at the variables, let’s first focus on 𝑥. So we can see that the smallest power of 𝑥 is 𝑥 squared, so 𝑥 squared goes into both of them. And the smallest power of 𝑦 is 𝑦 squared, so 𝑦 squared goes into both. So our greatest common factor is a little bit more complicated this time; our greatest common factor is six 𝑥 squared 𝑦 squared, so we gonna do exactly as we did in the previous two examples. And we’re gonna take each of those terms and write them as multiplications with the greatest common factor. So as always we do the numbers first, so we can say that six multiplied by two is twelve.
Then looking at the 𝑥 squared term, we can see that 𝑥 squared multiplied by one is 𝑥 squared, so we don’t need to do anything with that. But we know that 𝑦 to the power of five is 𝑦 multiplied by 𝑦 multiplied by 𝑦 multiplied by 𝑦 multiplied by 𝑦. And we know that 𝑦 squared is just 𝑦 multiplied by 𝑦, so we can see that we’ve got three left over 𝑦s. So we’ll need to multiply our term by 𝑦 to the power of three or 𝑦 cubed. Now so we don’t get confused, we’re going to take this next term as all of negative thirty 𝑥 to the power of four 𝑦 squared. So six multiplied by negative five is negative thirty.
And then looking at the 𝑥 powers, we can see we’ve got 𝑥 squared in our greatest common factor. And we need to get two 𝑥 to the power of four. So we know that 𝑥 to the power of four is 𝑥 multiplied by 𝑥 multiplied by 𝑥 multiplied by 𝑥. And we know that 𝑥 squared is just 𝑥 multiplied by 𝑥. So we can see that we’ve got two left over to get to our term, so that will be negative five 𝑥 squared.
Now we’ve done all we need to do. So we’re going to use the distributive law again by taking our greatest common factor and putting that on the outside of the parentheses. And then what we got left over is two 𝑦 cubed minus five 𝑥 squared and there we have it. So just to recap what we did, we went to a polynomial and we looked for the greatest common factor first by looking at the constants. So we said what of twelve and negative thirty; what’s their greatest common factor? And we can see that that was six. Then we looked at the variables and found the lowest power of each one; that was 𝑥 squared and 𝑦 squared. We then looked at each term individually and said what do I have to multiply this greatest common factor by to get the term above. So we found that that we’ll use the distributive law, putting that on the outside of the parentheses and leaving us with two 𝑦 cubed minus five 𝑥 squared.