In this video, we're going to talk about how to add, subtract, and multiply polynomial expressions. So let's begin. Let's say if we have 4x squared plus 5x plus 7 plus 3x squared minus 8x plus 12. So how can we add these two polynomial expressions?
If you know what to do, feel free to pause the video and work out this particular example. What we need to do is combine like terms. 4x squared and 3x squared are like terms.
So let's add them. 4 plus 3 is 7, so this is going to be 7x squared. Now 5x and negative 8x are like terms. 5 minus 8 is negative 3. And finally we can add 7 and 12, which together is 19. So that wasn't too bad right? Let's try another example.
Go ahead and try this one. 9x squared minus 7x plus 13 minus 5x squared minus 7x and minus 14. So go ahead and subtract these two polynomial expressions. Now the first thing I will do is distribute the negative sign to every term on the right. The signs will change.
On the left side, you can just open the parentheses. If there's no number in front of it, you can just rewrite it as 9x squared minus 7x plus 13. And then if we distribute the negative sign to the other three terms, it's going to be negative 5x squared plus 7x plus 14. And now let's combine like terms. So we can combine those two.
9 minus 5 is 4, so it's 4x squared. Negative 7x plus 7x is 0, so they will cancel. And 13 plus 14 is 27. So this is the answer, 4x squared plus 27. So here's another problem that we can work on. 3x cubed minus 5x plus 8 minus 7x squared plus 6x minus 9. So let's distribute the negative sign, just like we did before.
So the first three terms will remain the same. And then we'll have negative 7x squared minus 6x plus 9. So now let's go ahead and combine like terms. So there's no similar term to 3x cubed.
There's only one x cubed term. So we're just going to bring it down and rewrite it. Likewise, this term is one of a kind.
So we're just going to rewrite it. Now we can combine these two terms. Negative 5 minus 6 is negative 11. And 8 plus 9 is 17. So this is the answer. 3x cubed minus 7x squared minus 11x plus 17. Now what if we had numbers in front?
what would you do in this case so the first thing we should do is distribute the 4 to these three terms so 4 times 3x squared is 12x squared And then 4 times 6x, that's equal to 24x. And 4 times negative 8 is negative 32. Now let's distribute the negative 3 to the three terms on the right. Negative 3 times 2x squared is negative 6x squared.
Negative 3 times negative 5x is positive 15x. And finally, negative 3 times 7 is negative 21. So now let's combine like terms. 12 minus 6 is positive 6. 24 plus 15 is 39. Negative 32 minus 21 is negative 53. So this is it.
Now let's talk about how to multiply polynomial expressions. Let's start with two binomials. So let's say if we have 3x plus 5 multiplied by 2x minus 3. We need to use the Fourier method.
3x times 2x is 6x squared. 3x times negative 3 is negative 9x. 5 times 2x is 10x.
And finally, 5 times negative 3 is negative 15. So now at this point, we can combine like terms. Negative 9 plus 10 is positive 1. The other two terms, we can bring it down. So it's going to be 6x squared plus 1x minus 15. So that's what you can do in order to multiply two binomials together.
Now what if you were to see an expression that looks like this? 2x minus 5 squared. How can you simplify this expression?
If you see something like this, this simply means that you have two binomials multiplied to each other. So there's two 2x minus 5s. So let's do what we did in the last example.
Let's FOIL. 2x times 2x. is equal to 4x squared.
2x times negative 5 is negative 10x. Negative 5 times 2x is also negative 10x. And finally negative 5 times negative 5 is positive 25. So now let's combine these terms.
Negative 10x minus 10x is negative 20x. And so this is the answer. It's 4x squared minus 20x plus 25. Now what if we want to multiply, let's say, a binomial by a trinomial? How can we do so?
Now notice that when we multiply a binomial with another binomial, that is, an expression with two terms by another expression with two terms, initially we got four terms before we added like terms. Now in this example, we have a binomial which contains two terms and a trinomial which has three. 2 times 3 is 6. So when we multiply, before we combine like terms, we should have six terms. So let's go ahead and multiply.
4x times x squared is 4x cubed. 4x times 3x. is 12x squared.
4x times negative 5 is negative 20x. Negative 2 times x squared is negative 2x squared. Negative 2 times 3x is negative 6x. And negative 2 times negative 5 is positive 10. So let me just double check and make sure that I didn't make any mistakes. So I believe everything is good.
Now let's go ahead and combine like terms. It's always good to double check your work. So this term is one of a kind. So let's simply rewrite it. These two are like terms.
12 minus 2 is 10. And these two are like terms. Negative 20 minus 6 is negative 26x plus 10. But as you can see, before we combine like terms, notice that we have a total of 6 terms initially. Anytime you multiply a binomial by a trinomial, you will initially get 6 terms. Now what's going to happen if we multiply a trinomial by another trinomial? Go ahead and try it.
So 3 times 3 is 9. Initially, before we combine like terms, we should have 9 terms. So 3x squared times 2x squared is 6x to the 4th power. And then 3x squared times 6x, that's going to be 18. 3 times 6 is 18. x squared times x is x cubed.
And then 3x squared times negative 4 is simply negative 12x squared. Next, we have negative 5x times 2x squared. Negative 10x cubed, and then negative 5x times 6x, which is negative 30x. And negative 5x times negative 4. Wait, negative 5x times 6x is negative 30x squared. It's always good to double check the work.
Negative 5x times negative 4 is 20x. And then 7 times 2x squared, that's going to be 14. x squared and then 7 times 6x is positive 42x and finally 7 times negative 4 is negative 28 so I'm just going to take a minute and double check everything make sure I didn't miss anything So I believe everything is correct up to this point. So as you can see, we have nine terms at this point.
Now let's go ahead and combine like terms. So we have 6x to the 4th, and we can combine these two. 18 minus 10 is positive 8. And there's three terms with an x squared attached to it.
negative 12 plus 14 is positive 2 and positive 2 minus 30 is negative 28 Now we have these two terms to add. 42 plus 20 is 62. And then the last term. So this is it. 6x to the 4th plus 8x cubed minus 28x squared plus 62x minus 28. So now you know how to multiply a trinomial with another trinomial. Now what about dividing polynomials?
Let's say if we wish to divide the trinomial x squared plus 7x plus 15. Actually, instead of plus 15, let's say plus 12. Let's divide it by x plus 3. How can we do so? There's three things that you can do. You can factor, you can use long division, or you can use synthetic division.
Let's divide by factoring. To factor the trinomial, we need to find two numbers that multiply to 12, but add to 7. 3 times 4 is 12, 3 plus 4 is 7, so we can factor it like this. It's x plus 3 times x plus 4. Now we can cancel these two terms.
So therefore, it's x plus 4. So x squared plus 7x plus 12 divided by x plus 3 is x plus 4. So that's how you can divide two polynomial expressions by factoring. Just factor and cancel. Now let's try another example.
2x squared minus x plus 6 divided by x minus 2. Now you can factor the numerator. It is factorable, and you can cancel it, so you can use the other method as well. But for this particular example, let's use long division.
So I'm going to put the denominator on the outside and the numerator on the inside. So first, we're going to divide 2x squared by x. 2x squared divided by x is 2x. Now we're going to multiply. 2x times x is 2x squared.
And 2x times negative 2 is negative 4x. And now subtract. 2x squared minus 2x squared is 0, so those two cancel. And then negative 1x minus negative 4x is the same as negative 1x plus 4x, which is positive 3x. 6 minus nothing, or 6 minus 0, is simply 6, so we can bring the 6 down.
Now let's try another example. Let's divide 2x squared minus 7x plus 6 by x minus 2. Now the numerator is factorable, but we're going to use synthetic division and long division. You can factor and cancel if you want, but let's start with long division.
Let's put the denominator on the outside. and the numerator on the inside. So first, let's divide. 2x squared divided by x is simply 2x. So now let's multiply.
2x times x is 2x squared. 2x times negative 2 is negative 4x. And now we're going to subtract. 2x squared minus 2x squared is 0. They cancel.
Negative 7x minus negative 4x, which is the same as negative 7x plus 4x. That's negative 3x. And 6 minus nothing, or 6 minus 0, is simply 6. So we can bring the 6 down.
So now let's divide. Negative 3x divided by x is negative 3. And now let's multiply. Negative 3 times x is negative 3x.
And negative 3 times negative 2 is positive 6. So now let's subtract. Negative 3x minus negative 3x, or negative 3x plus 3x is 0. 6 minus 6 is 0, so the remainder is 0. Therefore, this is equal to 2x minus 3. So that's how you can divide polynomial expressions using long division. Now let's see if we can get the same answer using synthetic division. let's write the coefficients of the numerator which are 2 negative 7 and 6 now we're dividing it by x minus 2 If you set this equal to 0, x is 2. So we're going to use 2 here instead of negative 2. Let's bring down the 2. 2 times 2 is 4. And negative 7 plus 4 is negative 3. So you've got to multiply, add, multiply, add, and so forth.
2 times negative 3 is negative 6. And 6 plus negative 6 is 0. So this is the remainder. Negative 3 is the constant, and 2 has the x with it, so it's 2x minus 3. When you divide 2x squared by x, you're going to get 2x. So the first term is x to the first power.
So you can divide polynomials by factoring, by using long division, or synthetic division. So that is it for this video. Thanks for watching.
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