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Adding and Subtracting Polynomials

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Today we're gonna talk about adding and subtracting polynomials. So we'll start by reviewing how to combine like terms within a polynomial, and then we'll do some examples adding and subtracting polynomials.

So let's start by reviewing combining like terms. I have this polynomial x to the third plus 2x minus 8x plus 6. I know that I can combine terms if they have the same variable and the same power. So the first term has a variable of x and a power of 3. It also doesn't have a coefficient, but it is still considered a term because a term without a coefficient is implied to have a coefficient of 1. We just don't add it. We just don't write it. So I want to see if I can combine this term with any other term. But no other term has a power of 3, so we can't combine this with anything.

My next term is 2x which has a variable of x and a power of one. And I see that I have another term with the same variable x, and it's also the same power of one. So these two terms are alike, and I can combine them. My last term is six which is just a constant. And I have no other constant terms, so I will not be able to combine this with anything.

So I'm gonna go ahead and combine my two middle terms. I'm going to combine them either with adding or subtracting. I'm going to add if the coefficient in front of the variable is positive, and I'm gonna subtract if the coefficient in front of the variable is negative. So to combine these, I'm gonna go ahead and combine positive 2 minus 8. That's gonna give me a negative 6x. I can't combine my first term or my last term, so I'm just gonna bring those down. And my simplified expression is x to the third minus 6x plus 6. So we combined our two terms, very similar to the way that we would add or subtract monomials, to arrive at this simplified expression.

So here's another example of combining like terms. I've got this polynomial, and I'm going to identify which terms I can combine, or which terms are like terms, by looking at which ones have the same variable and power. So my first term, it has the variable and power x squared y to the fourth. And I have another term that has an x squared but not a y to the fourth. So because these two terms do not have the same exact exponential associated with the same variables, I cannot combine them.

I can however combine this first term with this term, positive 2 x squared y to the fourth, because again they both have an x squared and a y to the fourth. So 3x squared y to the fourth plus 2x squared y to the fourth is going to give me 5x squared y to the fourth. And I can see that I am adding because my coefficient is positive.

I then have a negative 5y and a positive 2y. So because these have the same variable and the same power of one, we can combine them. So I have a minus sign in front of my term, so this is like a negative 5 plus a positive 2y. Negative 5 plus 2 is going to give me a negative 3y. So I'll use the minus sign because my coefficient is negative.

I couldn't combine-- whoops, I combined these two terms. I could not, however, combine this term with anything. So I'm just gonna bring this term down at the end. So now I have a simplified form of my polynomial that I reached by combining like terms.

So here's my first example. I'm gonna combine two polynomials through addition. So I'm gonna start by writing my polynomials in standard form and using a zero coefficient for any term that's missing. So this first polynomial will become 4x to the third plus 6x squared plus 0x plus 2.

|I'm gonna do the same thing for my second polynomial and again use the zero coefficient for any term that's missing. I don't have a term with an x to the third so I'll start with 0x to the third. I have a 2x squared. I have a negative 3x. And I have a negative 4 for my constant term.

So now I can combine these by adding. And it'll be easier to combine my like terms because they are lined up vertically so 4 plus 0 is going to give me 4x to the third. 6x squared plus 2x squared will give me positive 8x squared. 0 plus negative 3 will give me minus 3. And then my x. And 2 plus a negative 4 will give me a minus 2. So these two polynomials added together will give me this for my final answer.

For my second example, I'm gonna combine two polynomials by subtracting. So I'm going to start this in the same way that I did my first example, by writing each term that's missing with a zero coefficient. So I have my first term as negative 5x to the fourth. And then I'm missing a term with x to the third so I'll write this as 0x to the third. Then I'll have my plus 3x squared. I'm missing my term with just x, so I'll have plus 0x. And then I'll bring down my plus one.

For my second polynomial, I'm going to be subtracting 2x to the third, so I'll line that up with my 2x to the third. And because I don't have an x to the fourth term, I'll write this as 0x to the fourth. Then I have a 2x term, so I'll write that underneath my other x term. Again, because I don't have an x squared term, I'll write this as 0x squared. And finally I have my plus 6.

Now I'm going to put this whole polynomial in parentheses so that I know that I'm subtracting all of the terms in this second polynomial. And I'm going to subtract down vertically. So negative 5 minus 0 will give me negative 5 for my coefficient, and I'll bring down my x to the fourth. 0 minus positive 2 is going to give me negative 2 for my coefficient, and I'll bring down x to the third. 3 minus positive 0 or 3 minus 0 is just going to give you a 3x squared. 0 minus 2 is going to give me negative 2, and I bring down my x. And finally, 1 minus 6 is gonna give me negative 5. And so now I have this polynomial which is the combination of these two polynomials through subtraction.

So let's go our key points from today. You can combine terms using addition or subtraction if the terms have the same variables and associated variable power. A term with a subtraction sign before it signifies that the coefficient for the term is negative, and a term with an addition sign or no sign indicates that the coefficient for the term is positive. And when combining like terms, add the coefficients and keep the variables and powers the same.

So I hope that these key points and examples help you understand a little bit more about adding and subtracting polynomials. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.