in this video we're going to do a basic review of simplifying algebraic expressions so let's say if you have the expression 5x + 8 - 2x + 5 how can we simplify this expression to do so you need to combine like terms we can't add 5x and 8 we can't say that's 13x because these are not like terms 5x is similar to 2x because they carry the same variable X so we can combine 5x and -2X 5 + -2 is the same as 5 - 2 and that's 3x now 8 and five are constants they're like terms so we can add those two numbers since they don't have a variable 8 + 5 is 13 and so the answer for this problem is 3x + 13 so this expression is equivalent or it has the same value as this expression here let's try another problem simplify the following expression 3x + 5 y - 9x + 7 y so first let's identify the terms that are similar to each other so 3x and 9x are like terms 5 Y and 7 y are like terms they're similar so let's combine 3x and 9x 3 - 9 is -6 5 + 7 is 12 so this is the answer -6x + 12 y or we can write it like this as well 12 y - 6X these expressions are equivalent expressions here's another problem that you could work on simplify the following expression 5x y plus 6 x^2 + 8 x y - 9x2 so feel free to pause the video and work on this particular example so let's begin by identifying which terms are light terms 5xy and 8xy are like terms they both have the variables X Y so 5 + 8 is 13 so it's going to be 13x Y and 6 x^2 9x2 they're like terms 6 - 9 is3 so it's going to be -3x 2 so this is the answer now before we move on another concept that you need to be familiar with is the distributive property so let's say if we have a 9 mtip 5x + 4 to simplify this expression we can distribute 9 to everything inside the parentheses so 9 * 5x is 45x and 9 * 4 is 36 so this expression is equal to 4 45x + 36 let's try another example using in the distributive property so try this one 7 * 3x^2 - 8x + 2 so let's distribute 7 * 3 is 21 the X2 will carry over 7 * -8x is 56x and 7 * pos2 is pos4 so that's how you can use the distributive property to simplify Expressions try this one simplify this expression 5 * 3x + 4 - 7 x + 8 so go ahead and simplify this first let's distribute 5 * 3x is 15x and 5 * 4 is 20 and let's rewrite what we have on the right -7x + 8 so now let's combine like terms so let's combine 15x and -7x 15 - 7 is 8 and we can combine the constants 20 and 8 20 + 8 is 28 So This Is The Answer let's work on another example let's try stimes 2x - 3 + 4 * 9x - 5 go ahead and work on this example simplify this expression distribute and then combine like terms so 7 * 2x is 14x and 7 * -3 is -21 pos4 * 9x is POS 36x pos4 * -5 is -20 whenever you multiply two positive numbers you will get a positive result if you multiply a positive number and a negative number you're going to get a negative number and whenever you multiply two negative numbers you're going to get a positive number so make sure to remember those rules now let's combine like terms so 14x + 36x is equal to 50x -21 + -20 which is the same as -21 - 20 that's -41 so this is the answer to this particular problem so that's all you need to do let's try another example 3x^2 + 5x - 6 - 4x^2 + 7 x minus 3 so if you're taking a test and you see a problem that looked like that what would you do now the first thing that we can do is we can get rid of the parentheses on the left side we really don't need it so basically if you don't see a number in front it's an invisible one and the same is true here it's a negative one in front of it so one time everything inside of the parentheses will be whatever is inside of the parentheses 1 * 3x 2 is 3x^2 1 * 5x is 5X and 1 * -6 is6 so we really don't need the parentheses on the left because there's nothing in front of it except an invisible one now the negative sign is important for the parentheses on the right if we multiply everything inside the parentheses by 1 it will have an impact -1 * 4x^2 is - 4x^2 so the sign is going to change -1 * 7x is -7x and -1 * -3 two negative numbers that are multiplied to each other will give you a positive number so this is going to be positive three now let's combine like terms 3x^2 - 4x^2 is - 1x^2 or simply x^2 5x - 7x is -2X and -6 + 3 is -3 so the final answer is x^2 - 2x - 3 so that's how you can simplify a polinomial expression that looks like that so this is called subtracting pols so let's try another example 5 * 2x Cub + 5 x^2 - 8 - 3 * 4x^2 + 5x + 9 go ahead distribute and then combine like terms so let's multiply 5 by 2x Cub so that's going to be 10 x cub and then 5 * 5x^2 is 25 X2 5 * 5 is 25 and then 5 * 8 is -40 now let's distribute -3 to everything on the right side so -3 * 4 is uh -12 and -3 * 5 is-5 or -5x and -3 * POS 9 is -27 so now what we need to to do is combine like terms so we only have 1 x Cub we have uh 2 X2 one term with X and two constants so we can't combine 10 x Cub with anything so let's simply rewrite it in the next line so 10 x Cub now we can combine 25 x^2 and -2 X2 they're like terms 25 - 12 is 13 now there's nothing to combine the -5x with so it's by itself let's rewrite it and finally we can combine -40 and -27 which is -67 so this is the answer now what's going to happen if we multiply a monomial by another monomial if you multiply x^2 * X Cub this is going to be X to 5th power whenever you multiply a common variable you need to add the exponents so with that in mind try these examples X 4th * X 7th x 3r * X8 and X5 * X to Theus 3 so 4 + 7 is 11 3 +8 is5 by the way if you have a negative exponent you can rewrite it like this this is 1 / X POS 5 if you move the variable from the numerator to the denominator that's the bottom of the fraction the exponent will change sign from 5 to posi 5 now what about the next One X to the 5th power * X to the3 if we add 5 and3 that's the same as 5 - 3 that's two so the answer is x^2 what if you saw a problem like this 4xb * 5x^2 what would you do so if you were to see this on a test how would you answer it the first thing we need to do is multiply the constants 4 * 5 4 * 5 is equal to 20 and then multiply the variables X Cub * x^2 and that's x to the 5th power so let's try some more examples try these 3 x 4th * 7 x 5 let's rewrite that and then uh 8 x to the 3r * 5 x 6 and 4 X7 * 3 x 5th try those examples so 3 * 7 is equal to 21 and X to 4th * X the 5th is X to the 9th because 4 + 5 is 9 so now let's move on to the next example 8 * 5 is 40 x Cub * x 6 3 + 6 is 9 so that's X to 9 4 * 3 is 12 and 7 + 5 is also 12 so this is 12 x to the 12th power what about multiplying monomials with multiple variables try this one so the principle is the same let's multiply the constant first 8 * 7 is 56 x^2 * X Cub is X the 5th power since 2 + 3 is 5 y Cub * y 4th power is y 7 since 3 + 4 is 7 so let's try some examples try this one 5x Cub y 4 * 7 x^2 y 8 and then 3x y^ 2 * 5 x Cub Y and also 4 x to the 4 y -3 * 5 x 3r y the 5 so 5 * 7 is 35 x Cub * x^2 is X 5th power and for y 4 * y 8 4 + 8 is 12 so it's y 12th power now let's go ahead and make some space so let's get rid of that stuff and let's move this over here 3 * 5 is 15 now what is x * X Cub if you don't see an exponent there is an invisible one so it's really 1 + 3 so it's X 4th and here this is y^2 * y to the first Power 2 + 1 is 3 so that's going to be y Cub now let's try the last example so let's multiply 4 * 5 which is 20 and then X 4th * X 3r that's x to the 7th power and then y -3 * y5 -3 +5 is8 but since we have a negative exponent we can move it to the bottom so it's 20 x 7 / y 8 power so we talked about multiplying monomials we said that x 4 * X 5th is x 9 because 4 + 5 is 9 now what about dividing monomials let's say if we want to divide x 8 by X 3r if you're going to divide you need to subtract the exponents 8 minus 3 is 5 and that's basically what you need to do so let's uh work on some examples try these well actually let's just do this one first and then I'll give you some more so for this one it's basically 7 - 4 which is three so it's X Cub this is the answer now another way in which you can see it X to 7 is basically 7 x VAR multiply to each other X 4th is basically 4 x variables multiply to each other you can cancel 4 x's and notice that you'll be left with three x variables on top which is X to the 3 let's try this example x^2 / X 5th power so we need to subtract the top exponent by the bottom exponent so this is going to be a 2 Min - 5 which is3 and whenever you have a negative exponent you want to get rid of the negative sign so if the negative exponent is on top move the X variable to the bottom or to the denominator of the fraction so this is going to be 1/x Cub now another way in which you could see that is you can expand the expression X2 is basically x * x and x the 5th power is x * x * x 5 times basically now we can cancel 2 x variables so if you cancel everything and if there's nothing on top there going to be a one because x / X is one notice that we have three x variables on the bottom so it's 1X Cub as you can see the answer is the same so that's another way in which you could uh consider it when you're multiplying or dividing by monomials if you're multiplying binomials let's say x^2 * X Cub the reason why it's X 5th power is because x^2 is x * x x Cub is x * x x x so you have five x variables I meant to say five x variables it kind of rolled off my tongue in a wrong way but that's uh x to the fifth power as you can see so let's try some more examples try these X to 9 / X 4th X Cub over x to the 11th x-4 / X 5th x 3 / X8 and x^ -7 / X5 so let's start with the first one this is going to be 9 - 4 which is 5 so that's it for that example and now let's work on this one so this is going to be the top number minus the bottom number so 3 - 11 which is8 and we can rewrite that as 1 x 8 so now let's try this one so this is going to be -4 minus 5 the top number minus the bottom number you can put it in parentheses if that helps -4 minus 5 is9 and we can rewrite that as 1/ x 9 another way in which you could solve this problem if you want to see it in a different way you can move the X variable from the top to the bottom if you do that the ne4 will become pos4 so this is 1 / x 4 * X 5th and we know that X 4th * X 5th is X to the 9th because 4 + 5 is 9 so that's another way or a quick way to get the same answer so let's try this one this is going to be the top number three minus the bottom number which is8 and since we have two negatives next to each other this is the same as 3 + 8 which is 11 now another way in which we could solve it we can take the X variable move it to the top and so this is going to be positive x to the 8 or x to the positive 8 which will still give us x to the 11th so as you can see there's more than one one way to get the answer now for this one we can subtract the exponents the top minus the bottom -7 -5 which is the same as -7 + 5 which is -2 -7 + 5 is the same as 5 - 7 that's -2 which is 1 /x^2 so that seems like a lot of work or we can do it this way we can take the X to the 7 on the top move it to the bottom so that the exponent will be positive and the other X the X to the 5 we can move it to the top so this is X to the 5th / X to 7th x to the 5th is basically x * x * x five times and x to the 7 means that we have 7 x variables on the bottom so we can cancel five on top and five on the bottom which means that we have two left over on the bottom so x * X is X2 so it's 1/x squ that's another way in which you can get the answer now what if you have examples like this what's 36 x 7th / 4 x Cub how would you simplify this expression so first divide the constants 36 / 4 is 9 and then then focus on the variables x 7 / X Cub that's going to be 7 - 3 which is 4 and so that's the answer for that one try this one 28 x to 5th y 4th / 7 x to the 8 y to the let's say -3 so let's divide 28 by 7 which is four and then 5 - 8 is -3 that's for the X variable and for the Y variable it's U 4 - -3 which is 4 + 3 and that's 7 so it's Y 7th now since we have a negative exponent we need to move it to the bottom so this is 4 Y 7th / X Cube so that's the answer now what if you have a fraction with two terms in the numerator let's say something like this and you're dividing it by a monomial or a single term how can we simplify this expression if you ever see an expression like this you could separate the single fraction into two smaller fractions so this is equivalent to 40 x 5 / the 4X and 12 x s / 4X so divide each term on top separately by 4X so 40 / 4 is 10 and 5 - 1 is 4 so this is going to be 10 x to the 4th power 12 / 4 is 3 and 2 - 1 is 1 so it's 10 x 4 + 3x now let's try another example like that so try this one 36 x Cub + 18 X2 plus uh 15x / 3x so this is equivalent to 36 x Cub / 3x + 18 x^2 / 3x + 15x / 3x 36 / 3 is 12 and 3 - 1 is 2 so this is going to be 12 x^2 18 / 3 is 6 2 - 1 is 1 and the last term the X variables will cancel so it's simply 15 / 3 which is 5 and this is the answer so that's how you can simplify this expression that's what you can do whenever you dividing a polom which has many terms by a monomial which has one term now what if we wanted to multiply a binomial which contains two terms by another binomial how would you do it so this process is called foiling we need to foil oil or multiply the two binomials so let's multiply the first two terms on the outside 3x * 2x which is 3 * 2 is 6 x * X is x^2 because 1 + 1 is 2 3x * -3 is -9x 5 * 2x is 10 x and 5 * 3 is5 all we need to do now is add like terms so -9x + 10 x -9 + 10 or 10 - 9 is 1 1 x is simply X So This Is The Answer 6 x^2 + x - 15 now what if you have an expression that looks like this what would you do 5x - 4^ 2 is equivalent to 5x - 4 * 5x - 4 it simply means that you have two of them 5x * 5x is 25 x^2 5x * -4 is -2X -4 * 5x is -2X and finally -4 * -4 is POS 16 whenever you multiply two negative numbers you will get a positive number so now we can add the two terms in the middle -20 + -20 is -4x So This Is The Answer 25 x^2 - 40x + 16 now let's say if we wanted to multiply a trinomial a trinomial is an expression with three terms and let's multiply by a binomial which has two terms so 3 by two that should be six 3 * 2 is six so initially when we multiply these two expressions we should have six terms for the last example when we multiplied a binomial by a binomial it was a 2X two so initially we got four terms before we simplified it 3x^2 * 3x is 9 X Cub 3 * 3 is 9 x^2 * X the 1st it's going to be X Cub plus since 2 + 1 is 3 3x^2 * -7 is -21 x^2 2x * 3x is 6 x^2 and then 2x * -7 is -4x 4 * 3x is 12x and 4 * -7 is 28 so let's combine like terms -21 + 6 or 6 - 21 that's equal to5 x^2 -14 + 12 or 12 - 14 that's -2X and then everything else let's bring it down so it's 9x Cub - 15 x^2 - 2x - 28 so this is the answer now let's multiply a trinomial by a trinomial so 2x^2 - 7 x + 4 * 3x^2 + 5x + 7 so we're multiplying three terms by three terms initially we should get nine terms before we simplify or before we combine combine like terms so let's foil 2 * 3 is 6 x^2 * x^2 is x 4 now what's 2x^2 * 5x 2 * 5 is uh 10 x^2 * X is X cub and then 2x^2 * 7 is 14x -7x * 3x^2 that's - 21 x Cub -7x * 5x is -35 x^2 -7x * 7 is - 49x 4 * 3x^2 is 12 x^2 4 * 5x is 20x and 4 * 7 is 28 now I do need to make one small correction 2x^2 * 7 is 14 x^2 I put 14 x so just want to make sure that was corrected now 6 x 4th is is a term by itself there's no other like terms similar to it so we can rewrite it now we only have two terms with an X Cub that's 10 x cub and 21x Cub so 10 - 21 is1 14 x^2 - 35 x² that's - 21 + 12 X2 that's 9 x^2 - 49x and 20x is 29x and then finally plus 28 so this is the answer as you can see whenever you multiply a trinomial by a trinomial you should get nine terms 1 2 3 4 5 6 7 8 nine and then after that you can simplify by combining like terms