Algebra 2 Operations on Polynomials

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Today, we are going to be working on operations on polynomials.0002

And for this lesson, we are going to cover addition, subtraction, and multiplication.0005

Division is covered separately under another lesson.0010

OK, reviewing how to add and subtract polynomials: add or subtract by removing the parentheses and combining like terms.0013

And recall that like terms, or like monomials, have the same variables to the same powers.0024

So, like terms have the same variables raised to the same powers.0030

For example, 2x and 6x are like terms; y⁴ and 5y⁴ are like terms; or x²z⁵ and 8x²z⁵ are like terms.0034

They have the same variables raised to the same powers, and they can be combined.0053

So, talking about adding polynomials; what you need to do is add the like terms together.0064

These are monomials; and now we are working with polynomials.0070

For example, if I have 5x² + xy + 4z, and I need to add that to 6x² + 2xy - 6z,0073

for addition, you just simply move the parentheses; the signs within the parentheses stay the same.0090

So, remove the parentheses, and combine like terms: I have 5x² and 6x², so that is going to give me 11x².0095

xy and 2xy gives me 3xy; 4z - 6z is -2z; that is addition.0114

Subtraction--you just have to be careful with the signs.0124

For subtraction: 3x² + 2xy + 5y, for example, minus 2x³...let's see...- 4x + 6y.0127

OK, now, for this first one, I can just remove the parentheses, because there is no negative sign in front of this, so I just take the parentheses away.0149

Now, when there is a negative sign, in order to remove the parentheses, make this addition;0157

and we are going to apply the negative sign to each term within the parentheses.0162

So, 2x³ becomes -2x³; -4x...if I take a negative and a negative, I will get + 4x.0167

If I take the negative and apply it to 6y, that is going to give me -6y.0179

Now, that allowed me to remove the parentheses; but I had to reverse the signs within the parentheses where the negative sign was applied.0184

OK, now, combining like terms: if I have 3x³ - 2x³, that is going to give me just x³.0192

2x + 4x is 6x; 5y - 6y is -y; so again, just be careful when you are removing the parentheses,0199

when there is a negative sign in front; you have to apply that negative sign to each term within the parentheses.0211

OK, when multiplying polynomials, you need to use the distributive property.0218

Recall that the distributive property states that a times (b + c) equals ab + ac.0224

There can be more than two terms in here, and the distributive property would apply in that case, as well, to give you ab + ac + ad, and so on.0233

So, let's look at multiplying a monomial by a polynomial, for example.0246

If I have 3x² times (2x³ + x - 5), I am going to use the distributive property right here;0251

and I am going to multiply 3x² times 2x³, + 3x² (now I am going0261

to apply that to this term) times x, and then 3x² again, times -5.0274

And this is going to give me 6x...and then, since I am going to be multiplying, I need to add the exponents.0285

So, these have the same base of x, but I need to add the exponents; that is going to give me 2 + 3; that is going to be x⁵.0297

OK, now this is going to be 3x² times x, so this is 2 + (this really is a) 1, so that is x³.0306

And then, 3 times -5 is -15x²; and then, I am writing this out as 6x⁵ + 3x³,0321

and then I am just going to make this - 15x².0330

And I can't go any further, because I cannot combine terms, since these are the same variable, but they are raised to different powers.0336

So, I can't simply combine those.0343

Looking at the first example, this is simply adding a polynomial to another polynomial.0347

And I am going to remove the parentheses; and since it is addition, I don't have to worry about any sign changes.0353

I just take away the parentheses and keep the signs the same.0358

So, remove the parentheses, and then combine like terms.0364

This is going to give me -3x² + 8x², is going to be 5x².0367

-4x and -3 is -7x; 7 minus 8 is -1.0374

So, this is very straightforward: just remove the parentheses and combine like terms.0383

Example 2 is asking me to simplify; and this time, it is subtraction, so I have to be a little more careful with removing the parentheses.0389

The first one does not have a negative sign in front of it, so I simply take out the parentheses.0397

Here, I am going to change this to a plus; but in order to do that, I have to apply the negative sign to each term within the parentheses.0403

A negative and a negative is going to give me positive 2x³.0413

A negative and a negative here is going to give me + 8y.0420

And then, a negative and a positive here--that is just going to be -y².0424

Now, I have gotten rid of the parentheses; I can combine like terms.0431

I have only one y³, so that is going to just stay by itself; this is 9y³.0435

Let's look for y² terms: well, I have -3y², and I have -y²; that is going to be -4y².0450

That is my y² terms; now I have a constant right here, positive 7.0464

And I am looking, and I don't see any other constants; so that is just going to stay by itself.0472

And then, over here, I actually have an x³ term, so this is going to also remain as it is, because there is nothing I can combine it with.0480

So, this is going to give me 9y³ - 4y² + 2x³...let's put the constant at the end here.0493

So again, the important point here is that, when you have a negative sign,0504

and you are applying it, you need to make sure that you reverse the sign for each term inside the parentheses.0508

OK, Example 3: we are multiplying a monomial by a polynomial, and we are going to use the distributive property.0514

That states that a times (b + c) equals ab + ac, and we can apply that to a third or additional terms, as well, inside the parentheses.0524

So, this is going to give me 4a²b times -5a³b²,0535

plus...again, I am multiplying 4a²b times the next term, 6a²b²,0543

and then, times the third term, 4a²b times -4ab³.0550

OK, so I am multiplying each of these out--multiplying 4 times -5 gives me -20; recall that, if exponents have the same bases,0557

I am going to multiply by adding the exponents (this rule).0568

It is a^m times aⁿ equals a^{m + n}.0572

So, this is going to give me 2 + 3, so that is a⁵; b...this is really a 1 implied, and 1 and 2 is 3: b³.0579

Here, I have 6 and 4, so that is 24; a² times a²...2 + 2 would be a⁴.0594

b to the first power, times b to the second...1 + 2 gives me b raised to the third power.0603

4 and -4 is -16; a² times a is really 2 + 1, so that is going to give me a³.0610

And then, b (really to the first power) times b³ is b⁴.0621

Now, I am looking to see if I have any like terms; and I don't.0631

There are no terms here where it is the same bases to the same power; if there were, I would combine them to complete my simplification.0634

But none of these are like terms, so I am done simplifying this expression.0641

OK, again, I am asked to simplify; and this time, I am multiplying a monomial times a binomial times a binomial, so it is a little more complicated.0647

And you could go about it by multiplying this...applying it to each term first;0654

or what I am going to actually do is multiply the binomials out and then work with this.0658

So, I need to apply the distributive property more than once here.0663

I am going to start out with this, keeping the 4z², and now multiplying out using the distributive property.0669

And recall: with two binomials, you can use the FOIL method.0678

So, multiplying the first terms is going to give me 15z².0681

Now, the outer terms--that is 12z; the inner terms: -35z; and finally, the last terms, multiplied by each other, are -28.0687

I can simplify a bit more, and this is 15z²; I have like terms in here, and 12 - 35 is -23z, minus 28.0703

OK, so I applied the distributive property in here, multiplying each term in one times each term inside the other parentheses.0717

Now, I need to apply the distributive property, again, to work with the 4z².0726

And that is going to give me 4z² times 15b² plus 4z² times -23z, plus 4z² times -28.0732

OK, so I get 4 times 15; that is 60; remember that a^m times aⁿ equals a...and then I add the exponents.0750

So, I am adding 2 and 2 to get 4; this is z⁴.0761

-23 times 4 is -92; z² times z to the first power is...2 + 1 is 3.0767

Now, 4 times -28 is -112; and here, I just have a z²--no z term there.0780

OK, so I am looking here to see if I can simplify further, and I cannot, because there are no like terms.0788

So, today we covered operations on polynomials, focusing on addition, subtraction, and multiplication.0795

And next time, we will talk about division with polynomials.0802

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