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Factor Polynomials with a Common Factor
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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.”
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So we’re looking for a greatest common factor that goes into both six 𝑥 and fifteen.
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Well we know that they’re both in the three times table; so three goes into them and there isn’t a bigger number.
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So we’ll write both of these as with a factor of three.
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So I could say six 𝑥 is three multiplied by two 𝑥 and fifteen is three multiplied by five.
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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.
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And then on the inside, we’ll have what’s left over from each term.
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We’ll have first of all two 𝑥 and then five.
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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.
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Now we must factor the expression seven 𝑥 cubed plus fourteen 𝑥 squared minus thirty-five 𝑥.
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So again what we’re looking for is a greatest common factor of each of those terms.
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So looking at the numbers first of all, we can see that each of those numbers are in the seven times table.
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So the greatest common factor is seven first of all.
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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.
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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.
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The next term, we know that seven multiplied by two is fourteen.
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So we’ve got seven 𝑥 multiplied by two first of all.
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And then focusing on the variables, we know that 𝑥 multiplied by 𝑥 is 𝑥 squared.
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So it’ll be seven 𝑥 multiplied by two 𝑥.
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And now our last term, so let’s look at it as a whole term of negative thirty-five 𝑥.
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And we know that seven multiplied by negative five is negative thirty-five.
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So now we have seven 𝑥 multiplied by negative five.
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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.
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And then on the inside, we’ve got each of those terms in blue.
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So first of all, we have 𝑥 squared plus two 𝑥 minus five.
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And now we have it; we have fully factored this polynomial.
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So just as a recap of what we did.
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We looked at each of the terms and we tried to find the greatest common factor.
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So we looked at the numbers first and we found that seven was common in each of the terms.
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And then looking for the variables, we could see that 𝑥 was common in each of the terms.
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So the greatest common factor was seven 𝑥.
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Then we looked at each of the terms individually and said seven 𝑥 multiplied by what is that previous term.
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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.
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Now let’s look at one with not just 𝑥, but also 𝑦.
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So we have to factor twelve 𝑥 squared 𝑦 to the power of five minus thirty 𝑥 to the power of four 𝑦 squared.
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So we’re looking first of all for the greatest common factor in numbers.
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So we can see that two goes into both; well, that’s quite small.
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So maybe there’s a larger one, but we know three goes in.
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Then if two and three goes in, then that means that six must go in.
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And there isn’t a bigger number than that.
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So we’ve got the greatest common factor as six.
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Now looking at the variables, let’s first focus on 𝑥.
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So we can see that the smallest power of 𝑥 is 𝑥 squared, so 𝑥 squared goes into both of them.
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And the smallest power of 𝑦 is 𝑦 squared, so 𝑦 squared goes into both.
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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.
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And we’re gonna take each of those terms and write them as multiplications with the greatest common factor.
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So as always we do the numbers first, so we can say that six multiplied by two is twelve.
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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.
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But we know that 𝑦 to the power of five is 𝑦 multiplied by 𝑦 multiplied by 𝑦 multiplied by 𝑦 multiplied by 𝑦.
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And we know that 𝑦 squared is just 𝑦 multiplied by 𝑦, so we can see that we’ve got three left over 𝑦s.
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So we’ll need to multiply our term by 𝑦 to the power of three or 𝑦 cubed.
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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.
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So six multiplied by negative five is negative thirty.
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And then looking at the 𝑥 powers, we can see we’ve got 𝑥 squared in our greatest common factor.
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And we need to get two 𝑥 to the power of four.
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So we know that 𝑥 to the power of four is 𝑥 multiplied by 𝑥 multiplied by 𝑥 multiplied by 𝑥.
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And we know that 𝑥 squared is just 𝑥 multiplied by 𝑥.
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So we can see that we’ve got two left over to get to our term, so that will be negative five 𝑥 squared.
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Now we’ve done all we need to do.
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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.
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And then what we got left over is two 𝑦 cubed minus five 𝑥 squared and there we have it.
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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.
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So we said what of twelve and negative thirty; what’s their greatest common factor?
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And we can see that that was six.
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Then we looked at the variables and found the lowest power of each one; that was 𝑥 squared and 𝑦 squared.
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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.
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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.