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 five x plus seven plus three x squared minus eight x plus twelve 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 nine x squared minus seven x plus thirteen minus five x squared minus seven x and minus 14. so go ahead and subtract these two polynomial expressions now the first thing i would 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 parenthesis 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 five x squared plus seven x plus fourteen and now let's combine like terms so we can combine those two nine minus five is four so it's four x squared negative seven x plus seven x is zero so they will cancel and thirteen plus fourteen is uh 727 so this is the answer 4x squared plus 27. so here's another problem that we can work on 3x cubed minus five x plus eight minus seven x squared plus six x minus nine 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 seven x squared minus six x plus nine so now let's go ahead and combine like terms so there's no similar term to three x cubed there's only one x cubed term so we're just gonna 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 four to these three terms so four 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 3 terms on the right negative 3 times 2x squared is negative x squared negative three times negative five x is positive fifteen x and finally negative three times seven is negative twenty one so now let's combine like terms twelve 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 three x plus five multiplied by two x minus three we need to use the foil method three x times two x is six x squared three x times negative three is negative nine x five times two x is ten x and finally five times negative three is negative fifteen so now at this point we can combine like terms negative nine plus ten is positive one the other two terms we can bring it down so it's going to be six x squared plus one x minus fifteen 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 two x minus five 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 fives 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 two times three is six so when we multiply before we combine like terms we should have uh six terms so let's go ahead and multiply 4x times x squared is 4x cubed 4x times 3x is x 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 26 x plus 10. but as you can see before we combine like terms notice that we have a total of six terms initially anytime you multiply a binomial by a trinomial you will initially get six terms 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 6 x to the fourth 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 that's negative 10x cubed and then negative 5x times six x which is negative thirty x and negative five x times negative four wait negative five x times six x is negative thirty x 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 six x to the fourth and we can combine these two eighteen minus ten is positive eight and there's three terms with an x squared attached to it negative twelve 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 fourth 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 seven x plus fifteen actually instead of plus fifteen let's say plus twelve let's divide it by x plus three 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 twelve but add to seven three times four is twelve three plus four is seven so we can factor it like this it's x plus three times x plus four now we can cancel these two uh terms so therefore it's x plus four so x squared plus seven x plus 12 divided by x plus three is x plus four so that's how you can divide two polynomial expressions um 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 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 two x times negative two is negative four x and now subtract two x squared minus two x squared is zero so those two cancel and then negative one x minus negative four x is the same as negative one x plus four x which is positive three x six minus nothing or six minus zero is simply six so we can bring the six down now let's try another example let's divide two x squared minus seven x plus six 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 two x times x is two x squared two x times negative two is negative four x 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 three x and six minus nothing or six minus zero is simply six so we can bring the six down so now let's divide negative three x 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 three x minus negative three x or negative three x plus three x is zero six minus six is zero so the remainder is zero therefore this is equal to two x minus three 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 two negative seven and six now we're dividing it by x minus two if you set this equal to zero x is two so we're going to use two here instead of negative two let's bring down the two two times two is four and negative seven plus four is negative three so you gotta multiply add multiply add and so forth two times negative three is negative six and six plus negative six is zero so this is the remainder negative three is the constant and two has the x with it so it's two x minus three 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 if you want to find more videos on algebra trig precal chemistry physics check out my website video.tutor.net or check out my channel um you can find my playlist on my website or on my channel so if you like this video feel free to subscribe and uh thanks for watching