in this video we're going to talk about how to add subtract and multiply polinomial Expressions so let's begin let's say if we have 4x^2 + 5x + 7 plus 3x^2 - 8x + 12 so how can we add these two polinomial 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^2 and 3x2 are like terms so let's add them 4 + 3 is 7 so this is going to be 7 x^2 now 5x and 8x are like terms 5 - 8 is3 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 9x2 - 7 x + 13 - 5 x^2 - 7x and -4 so go ahead and subtract these two two polinomial 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 parentheses if there's no number in front of it you can just rewrite it as 9 x^2 - 7 x + 13 and then if we distribute the negative sign to the other three terms it's going to be - 5x^2 + 7 x + 14 and now let's combine like terms so we can combine those two 9 - 5 is 4 so it's 4x^2 -7x + 7 x is zero so they will cancel and 13 + 14 is uh 7 27 So This Is The Answer 4x^2 + 27 so here's another problem that we could work on 3x Cub - 5x + 8 - 7 x^2 + 6 x - 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 -7 x^2 - 6X + 9 so now let's go ahead and combine like terms so there's no similar term to 3x Cub there's only one X Cub 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 terms5 6 is -1 and 8 + 9 is 17 So This Is The Answer 3x Cub - 7 x^2 - 11 x + 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 4 * 3x^2 is 12 x^2 and then 4 * 6 x that's equal to 24x and 4 * 8 is -32 now let's distribute the negative3 to the three terms on the right -3 * 2x^2 is - 6x^2 -3 * 5x is POS 15x and finally -3 * 7 is -21 so now let's combine like terms 12 - 6 is postive 6 24 + 15 is 39 -32 - 21 is 53 so this is it now let's talk about how to multiply polinomial Expressions let's start with two binomials so let's say if we have 3x + 5 multiplied by 2x - 3 we need to use the foral method 3x * 2x is 6 x^2 3x * -3 is -9x 5 * 2x is 10 x and finally 5 * -3 is5 so now at this point we can combine like terms -9 + 10 is POS 1 the other two terms we can bring bring it down so it's going to be 6 x^2 + 1X - 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 - 5^ 2 how can you simplify this expression if you see something like this this simply means that you have two binomials multipli to each other so there 2 2x - 5S so let's do what we did in the last example let's foil 2x * 2x is uh equal to 4x^2 2x * -5 is -10x -5 * 2x is also -10x and finally 5 * 5 is POS 255 so now let's combine these terms -10x - 10x is -2X and so this is the answer it's 4x^2 - 20x + 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 * 3 is six so when we multiply before we combine like terms we should have six terms so let's go ahead and multiply 4X * x^2 is 4X Cub 4X * 3x is 12 x^2 4X * -5 is -2X -2 * x^2 is -2X 2 -2 * 3x is -6x and -2 * 5 is pos1 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 -20 - 6 is uh -26 x + 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 now what's going to happen if we multiply a trinomial by another trinomial go ahead and try it so 3 * 3 is 9 initially before we combine like terms we should have nine terms so 3x^2 * 2x^2 is 6 x to 4th power and then 3x^2 * 6X that's going to be 18 3 * 6 is 18 X2 * X is X cub and then 3x^2 * -4 is simply -12 x^2 next we have -5x * 2x^2 that's -10 X cub and then -5x * 6X which is -3x and -5x * -4 wait 5x * 6X is -3x 2 it's always good to double check the work 5x * -4 is 20x and then 7 * 2x^2 that's going to be 14 x^2 and then 7 * 6X is positive 42x and finally 7 * -4 is -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 6 x 4th and we can combine these two 18 - 10 is postive 8 and there's three terms with an X squ attached to it -12 + 14 is POS 2 and pos2 - 30 is -28 now we have these two terms to add 42 + 20 is 62 and then the last term so this is it 6 x 4 + 8 x Cub - 28 x^2 + 62 x - 28 so now you know how to multiply a trinomial with another trinomial now what about dividing pols let's say if we wish to divide the trinomial x^2 + 7x + 15 actually instead of + 15 let's say + 12 let's divide it by x + 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 syntatic 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 * 4 is 12 3 + 4 7 so we can Factor it like this it's x + 3 * x + 4 now we can cancel these two uh terms so therefore it's x + 4 so x^2 + 7 x + 12 / x + 3 is x + 4 so that's how you can divide two polinomial expressions um by factoring just factor and cancel now let's try another example 2x^2 - x + 6 / x - 2 now you can Factor the numerator it is factorable and you can cancel see 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^2 by X 2x^2 / X is 2x now we're going to multiply 2x * X is 2X 2 and 2x * -2 is -4x and now subtract 2x^2 - 2x^2 is0 so those two cancel and then --1x - -4x is the same as -1x + 4x which is POS 3x 6 - nothing or 6 - 0 is simply 6 so we can bring the six down now let's try another example let's divide 2x^2 - 7 x + 6 by x - 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^2 / X is simply 2x so now let's multiply 2x * X is 2x^ 2 2x * -2 is -4x and now we're going to subtract 2x^2 - 2x^2 is 0 they cancel -7x - -4x which is the same as -7x + 4x that's -3x and 6 - nothing or 6 - 0 is simply six so we can bring the six down so now let's divide -3x / X is -3 and now let's multiply 3 * X is -3x and -3 * -2 is POS 6 so now let's subtract -3x - -3x or -3x + 3x is 0 6 - 6 is zero so the remainder is zero therefore this is equal to 2x - 3 so that's how you can divide polinomial 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 -7 and 6 now we're dividing it by x - 2 if you set this equal to zero X is 2 so we're going to use two here instead of -2 let's bring down to two 2 * 2 is 4 and -7 + 4 is-3 so you got to multip LLY add multiply add and so forth 2 * -3 is -6 and 6 + -6 is 0 so this is the remainder -3 is the constant and two has the X with it so it's 2x - 3 when you divide 2x^2 by X you're going to get 2x so the first term is X to the first power so you can divide polinomial 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 trade pre-cal chemistry physics check out my website video.net or check out my channel um you can find my playlists on my website or on my channel so if you like this video feel free to subscribe and uh thanks for watching