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Polynomials Divided by Monomials

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Today we're gonna talk about dividing polynomials by monomials. So we're gonna start by reviewing how to factor, and then we'll do some examples dividing polynomials by monomials. So let's start by reviewing factoring. Remember this example, I've got 15x plus 25. To factor I want to find the biggest factor that both 15x and 25 have in common, which is to look for the greatest common factor. So the biggest factor that both 15x and 25 share is going to be 5. So I'm going to start by writing 5. And then I'm going to write in parentheses whatever I need to multiply by 5 to give me 15x, plus whatever I need to multiply by 5 to give me 25. So, if I multiply 5 by 3x, 5 times 3 will give me 15, and then I'll have my x. Plus, if I multiply 5 by 5, 5 times 5 will give me 25. So the factored form of 15x plus 25, is 5 times 3x plus 5. Again, I can verify that by distributing. 5 times 3x gives me 15x, and 5 times 5 gives me 25. So I can see that I'm back to my original expression.

Let's try another example. I want to factor 4x to the third plus 12x squared. So again I'm gonna start by looking for the biggest factor that both 4x to the third and 12x squared have in common. So, if I start by looking at the coefficients, the greatest numerical factor that they share is gonna be 4. So I'm going to start by writing 4, and then in parentheses, I'll need to find what I should multiply by 4 to give me my original two terms. So to give me 4x to the third I'll need to multiply 4 by x to the third, or 1x to the third. And to get 12x squared, I'll need to multiply 4 by a positive 3x squared. Now this is not completely factored, because I can also factor something out of my variable components. Because x to the third is bigger than x squared, I can factor out an x squared from both terms. So this will become 4x squared in the front. To get 4x to the third, I'm gonna multiply 4x squared by another x. And to get 12x squared, I'm gonna multiply 4x squared by positive 3. So now I've factored out my biggest numerical factor, and my biggest variable factor. So this is in factored form, and I can verify that by distributing. 4x squared times x will give me 4x to the third, and 4x squared times 3 will give me 12x squared. So my expression is the same as my original expression.

So for my first example, I've got this polynomial divided by a monomial. So I'm gonna start by simplifying, and separating, the terms the numerator and the denominator to write three separate fractions. So this is gonna become negative 5x to the fifth over 5x, plus 10x to the third over 5x, minus 15x over 5x. Now to simplify each fraction, I'm gonna find the common factors between the numerator and the denominator to simplify and rewrite it in its simplest form. So, between negative 5 and 5, these both have a factor of 5. So this is gonna simplify to give me negative 1. Between x to the fifth and x, they both have a common factor of x. So this is gonna cancel out, and this will become x to the fourth. So my first fraction simplifies to be negative 1x to the fourth, or negative x to the fourth. My second fraction, my coefficients both have a common factor of 5. So this is going to cancel out, and this will become 2, 5 times 2 is 10. Then my second fraction, they both have a common factor of-- Sorry, for my variable component, they both have a common factor of x. So I can factor out an x, this will go away, and this is gonna become x squared. So this fraction simplifies to be a positive 2x squared. Finally, in my last fraction my coefficients, 15 and 5, both have a common factor of 5, so this will go away, and this will become three. And my two variables, x and x, both have a factor of x, meaning they cancel each other out. So this last fraction just simplifies to be minus or negative 3. I check that my polynomial's in standard form, and it is because the terms are in decreasing order of degree. So this is my final answer.

So for my second example, I've got the polynomial 9x to the third plus 3x squared plus 6, and I'm gonna divide it by the monomial 3x. If you're feeling pretty confident, go ahead and pause, try it on your own, and then check back and see if you've got the right answer. So I'm gonna solve this, or simplify it, in the same way that I did my previous example. I'm gonna separate this fraction into three separate fractions. So I'll have 9x to the third over 3x, plus 3x squared over 3x, plus 6 over 3x. Now I'm gonna simplify each of these fractions by looking for the greatest common factor between the terms in the numerator and in the denominator. So, looking at the coefficients of my terms, they have a common factor of 3. So this will cancel out, and this will become 3. Then x to the third and x have a common factor of x. So here this will also cancel, and this will become x squared. In my second fraction, both my coefficients are 3, so they both cancel each other out or become 1. The x part of my term, between the x parts of my two terms, the greatest common factor is x. So again, here this will cancel, and this will become x to the first, or just x. And finally here, my last fraction 6 and 3 have a common factor of-- greatest common factor of 3. So here this will cancel, and this will just become 2. And I don't have another x variable in my numerator. So this will become-- that will stay an x. So now rewriting this, I have 3x squared plus x plus 2 over x for my final answer.

So let's go over our key points from today. When dividing polynomials, you can write expressions as products of factors that may cancel out when dividing the expression in the numerator of a fraction by the expression in the denominator. Factors that cancel out must be the same in the numerator and in the denominator. And these factors are called common factors. So I hope that these key points and examples helped you understand a little bit more about dividing polynomials by monomials. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.