okay guys uh welcome back to Meson African motives we are still on our Mathematics Grade n and as you can see there's a typical exam question that we are going to have as a revision so that you'll be able to understand how do they ask questions on algebraic expressions uh how to simplify how to factorize uh which is something now that you understand from the introductions uh that we had so I'm not going to waste much of your time guys let us quickly rush through the questions this is question number three we are given 3.1 simplify the following expression so they were given uh Expressions that we supposed to work with so the first one on 3.1 one was to simplify this to add so we just adding whatever that you're given in this case so we given uh - 4x + 6 + 11 x - 5 so what are we supposed to do remember the idea of the like terms the issue of like terms we supposed to add or subtract the like terms the part that has got X there's X here there's X here these are like terms the constants together six and minus 5 these are the like terms so meaning to say this can be written as minus 4X plus 11x it's a plus this 11 it is carrying a positive so that's + 11 x then you've got plus a 6 and a minus 5 so take note we just rearranging of which you can just add this because you see your like terms so that's minus 4 + 11 so if we add - 4 + 11 you're going to get 7 but that will be 7 x then 6 - 5 that is a positive one so this is what you're going to have at the end so that was to simplify what is it that you given I'm supposed to add the given Expressions 3.12 again it is to simplify but then we are supposed to multiply uh these three which is multiplying the bracket of xus one there's a minus 4 which is multiplying the bracket of xus 2 remember what I said whenever you expanding the brackets you expand by the term or the number that is close that the number that that is in contact with the bracket the three is in contact with this bracket the minus 4 is in contact with this bracket the number just before just just before that bracket number just before the bracket the one that you consider so three affects this bracket you distribute throughout by three that's going to be 3 * X which is 3x 3 * -1 that is -3 you multiply again -4 multiplies this bracket that's -4 * X X which is -4x -4 * A4 Nega negative that's a posi 8 so as you can see at this stage you are back here just like what we had here the issue again is on like terms just like this part we back to this stage so back to the like terms so you can subtract 3x - 4x which is going to be - x you can just write 3x - 4X then we left with the constants which is the minus 3 and the positive 8 so this is - 3 + 8 3 - 4 that is -1 so it's supposed to have -1 x but -1 x is same as - x then we've got - 3 + 8 so - 3 + 8 this is going to give you a positive five that is you're going to obtain a positive five in the manner so that's it okay so you're supposed to be very very careful how to simplify the condition is given 3.13 calculate the value of we given this uh expression 2x^2 minus 4 if x is 3 if given that at a condition where X is three that means in place of X I'm going to substitute so this is 2 into 3 to the exponent of two minus a 4 that's it going to substitute in place of X so if you simplify this uh that is going to be two 3 to the exponent of 2 that's 3 * 3 which is 9 - 4 or you can just use your calculator 2 * 9 which is 18us 4 that was going to give you a 14 at the end or this stage after simplifying then you can write this stage then the answer or just from that stage the answer still you are going to have your marks there that's the idea two marks for that simplify again on 3.2 3.21 what is it that you're supposed to simplify we are given 3.21 5x being raised to the exponent of 3 * y * uh 2x y to the exponent of two like this being raised to the exponent of two uh and everything if you had to consider this all right sorry for that is being divided to 15 x to the exponent of 5 y to the exponent of 2 so what you need is to understand your exponent idea of the exponents how do we uh work out our laws of exponents remember that if you're given something of this nature x to the exponent of a to the exponent of B is same as X to the exponent of a if you are given an exponent and an exponent you simply multiply those exponents and I also said it can be they can be terms numbers under the inside the bracket being raised to a certain exponent let's say a everything is going to be affected so be m to the exponent of a everything will be affected by that so what am I trying to say with this idea you are going to see that there is this part the five x being raised to the exponent of it means everything is going to be affected by that exponent the five will be raised to the exponent of three the X will be also raised to the exponent of three so it was supposed to be 5 to the expon 3 x to the expon 3 but what is 5 to the expon 3 that is 25 5 * 5 times three times or you can just use your calculator that is 125 X to the exponent of 3 y take note we simplify this but there is a y which is multiply so there we're just going to multiply to this one so it's the same as 125 x Cub y this is being multiplied again if you check we've got the same consideration just like the previous case of an exponent and exponent I mean in a bracket being raised to a certain exponent so the two is going to be raised to the exponent of two which is going to give us a four 2 * 2 x to the exponent of 2 that is going to be X2 so take note here this is y to the exponent of 2 to the exponent of two again like that there's this two and there's that one that is outside so how do we treat an exponent and an exponent you multiply these two so that's going to be 2 * 2 which is 4 so this is going to give us y to exponent 4 and this whole part is being divided to 15 x to the expon 5 y to the expon of 2 so that's the idea so meaning to say in our simplification as long these stages can be presented we can now uh simplify in any way that we understand at this stage in any way that we understand all right considering the numbers in that case all right let us just consider this time the numerator on its own uh so that you understand me I'm just going to work what is on the numerator this part of the numerator on its own so that's 125 is going to multiply 4 125 * 4 that is going to give us a 500 this case the X and the X the mod Ling each other so what's the concept as long I'm working with the multiplication of the bases which are the same what am I going to do I'm going to add exponents the bases as long they are the same I have to add the expon there's X there's also X so I'm going to add so this will be x to the exponent of 3 + 2 which is a five same concept with the Y here this to the exponent of one so I'm going to add again 1 plus a four which is a five because the bases are the same so every part is being divided by 15 x to the expon of 5 y to the expon of 2 it's true I was supposed to actually use the laws in this case but these two guys they are the same exactly the same it's just like I'm dividing five and five you can just cancel so it's the same thing these two are the the same same exp same X same exponent so they can cancel all right so if you were to divide this 15 over 100 over 15 uh this 500 over 15 it was going to give you 100 over three so if you divide this on your calculator that's going to be uh 100 over three on this part all right let's just see this part together here so quickly on your calculator that is 500 over 15 so that's 100 over 3 so this part now I want you to see what is remaining here is the Y part this one which is dividing so the same concept like this one if you are dividing now this time you're dividing you are going to subtract the exponents as long the bases there are the same you subtract exponents so there we are dividing this is y this is also y so we going to subtract so this will be y to the exponent of 5us 2 which is 3 so that's it so that's what you're obtaining this is a multiplication this is just like over one so it's just like 100 y to the expon 3 over 3 * 1 which is a 3 like that that's how you can present it if you want you can just leave it like that so guys is as long you are dealing with algebra H you need to understand that the concept of our variables also with the concept of the laws of exponents together they work hand in hand they do work hand in hand so you have to be very very careful in each and every part that you uh simplify on 3.22 again we given to simplify uh but this time this is given as 2x + 4 y like this everything over x + 2 y then you'll be wondering what is it that I'm supposed to simplify this a fraction and I'm given an expression in the numerator here this an expression remember a combination of two terms a term and a term again an expression so in that case just check if there's anything common so that you can factorize we have anything common one there's two there is four what is the highest common factor of two and four so remember two is same as 2 to the expon of one four is same as 2 to the expon of two considering the the the highest common factor we take the one that is common but having the smallest exponent so meaning to say uh that is going to be a two 2 to the exponent of 1 is two so if I factor out two what am I going to have at the end perect out two here that is 2x / two the one that I affected so this will cancel I remain with X if I factor out a two here that means I've got plus 4 Y which is being divided to the two that I factored out so plus 4 y / 2 that means I'm going to have 2 Y and that is positive on the other expression that is in the denominator here we can see that there's nothing common and it's the same thing as this part that we having so what are you going to factor out a one factoring out a one does not affect anything because xid one that's x 2 y/ one that's 2 y so as you can see these two brackets are the same they can cancel out they are the same a over a is equal to one you can cancel out so as long you've got two terms that are the same two brackets that are the same two numbers that are the same you get a one so meaning to say got two over one which is a two so this whole part that we are having here this whole part was going to give us a two just like that just like that okay let's check the other part of the question factorize fully 3.31 x^2 - x - 6 we are supposed to factorize this trinomial 1 2 3 that's a quadratic expression which is a tral so remember that if the coefficient of x² considering that x s is a one like this I just have to open two brackets to factorize this I'm just going to need two brackets to obtain x s it means the first bracket I must have X the second bracket I'm I'm supposed to have X again so that x * X that's X2 so this is it guys I want you to understand this properly you are going to find the product of the constant which isus 6 you consider the product of the constant which is minus 6 so you need the product of -6 two terms the factors of -6 two product but these products or these two numbers that you need they are supposed to give you this term affecting X this number affecting X I mean which is minus one so the products of negative consider two numbers that you multiply to obtain a there are so many numbers minus 6 and one If I multiply this I get a ne6 the six and the minus one as long because one of the numbers there has to be a negative to get a negative six there can be a minus three and a two If I multiply this I get A6 there can be three and a minus two If I multiply this I get A6 all these numbers that you're seeing here if you multiply them they are going to give us A6 but the concept is now if we add these numbers where are we going to have the negative one this negative one so it's a trial and error you try okay these are your factors try to add them -6 plus a 1 you get a which is not this part so it's not this one it's totally out can't use this the same thing if you add this you're going to get a five which is not you go to this one let's add minus three I want you to see what's going to happen here minus 3 so this is a minus three plus a two those two terms that you said they give you a minus 6 those two numbers if you add the two numbers what you obtain minus one so if you check this is exactly which is giving us what the negative one so the concept is the two numbers must give us this product of it's a product two terms two numbers they must give you a Nega if you multiply the two numbers you must obtain a ne6 so the first thing just try the numbers that you multiply you get a ne6 but if you add each part if you add you must get this number affecting X so if you add this one you're going to get a positive one not a negative one so we need a part which when you add now those products you must get a 1 so it's the minus 3 and the two so this will be a minus 3 and a + two so that was uh the factorization part on this one so if it is an equation they will be asking you now is they'll put they just put a zero here and ask you to solve it's now an equation so this one is an expression you just factorize you're done all right so that was our 3.31 3.32 we are get gain given to factorize and this is 18 x^ 2us 200 so in this expression that we given we can see that it's a we are dealing with a binomial in this case there just two terms to be considered that's a binomial for a binomial the first thing that you can just consider is there is there a highest common factor that's it that's the first thing is there a difference of two squares those are the things that you're supposed to think of because they just two terms is there a highest common factor is there a difference of two squares so if you check this one is not a perfect square this one is not a perfect square the square root of 18 we can't determine the square root of 18 and have an exact value so that's not that's not that's not a difference of two squ so you test like thisare root of 18 this is not a whole number it's 3qu root of two it's a decimal so it's not a perfect square the square root of 200 10quot of two it's not a perfect square a perfect square is like this if you divide like I mean if you find the square root of 25 you get a whole number like this so 25 is a perfect square so meaning to say as we can see now this number or this expression is not a difference of two squares for now but if we factor out from 18 and 200 there is a highest common factor there the biggest number that we can factor out between 18 and 200 is what if you do not know this just express your numbers as product of their prime factors because you need to find the highest common factor so this is 18 so I can divide this side I can use this side it's up to me 2 into 18 or 18 2 that's 9 I can't use two I'm going to use three remember you're dealing with prime numbers so after two the prime numbers 2 3 5 seven those are the prime numbers one and itself divisible by one and itself so you're going to have 3 9/ 3 that's a three then a three 3/ 3 that's a one okay let's get to 200 we do the same thing too 200id two that is 100 can still use two 100 two that is a 50 we can still use two 50 divid two that's a 25 25 that's a five there we can use a five 25id 5 which is a 5 five and a one so if you do not know like because to find the common factor you just supposed to look at the numbers okay two is the common factor like just looking into the numbers but if you can't tell that express your numbers as the product of their prime factors the highest common factor is is a common number between these two between these two terms like if I can check here two is appearing here two is appearing here so two if I move on I've got three yeah I don't have three I've got three I don't have three two I no longer have two there's there's no there's no number that is left out which is common here it's only two which is common on these numbers so two is the highest common factor this one we no longer have a two already used it there's no longer a two there's no longer five there's no longer five the same concept on this side we do not have a three here we do not have a three here so a highest common factor must be common throughout so the common factor there was going to be a two so factoring out two we going to divide 18 / 2 which is 9 x 2 -/ 2 that was going to give us - 100 so before you conclude that this is the simplest part and underline your answer to say now I'm done test again am I having a difference of two squares Let's test again can we have are we having the square of n if you know the square roots by head then it's fine but if you do not know you're stuck in exam your calculator is there as your friend the square root of 9 that is a three meaning to say this is a perfect square the square root of 100 which is a 10 meaning to say this is a perfect square so meaning to say we are dealing or we are working with a difference of two squares on this part the difference of two squares remember a s minus b s is supposed to give us a minus B into a + b where these are perfect squares being raised to exponent of 2 so 9 is same as 3 2 x^ 2 - 100 which is - 10 2 so that's it so meaning to say I'm going to have two into the two brackets of the difference of two squares Aus B the a is the part that is being raised to the exponent of two so that's our 3x or the root of 9 which is 3 the root of X2 which is what x so this is our a minus and this is a plus then the B is the part being raised to the exponent of two on the other part which is 10 s so the B There is the 10 so we've got 10 and a 10 here or simply the square root of 100 so that's it so this is in simplest form there is nothing common if you are to consider on this part we do not have anything that we can say it is common that we can factor out again so the golden rule of the factorization the first thing you are supposed to factor out the highest common factor what is your highest common factor then check am I having a difference of two squares if there is a difference of two squares apply the difference of two squares if it's not there let's say I factored out two I'm left with x 2 minus 5 like this I'm done it's not a difference of two two squ this one five can I five is not a perfect square so if it is like that I'm done but in this case there was a difference of two squares so you have to apply the concept as it is given not because you just say after factoring out there's always a difference of 2 squ no and also to take note if it was a plus this cannot be factorized further 9x 2+ 100 that's a sum of two squares we can't factorize the sum of two squares so this will be done at this stage if it was like this you're going to just remain at that stage but this one can be done because there's a negative and these are perfect squares that makes it to be a difference of two squares that's the idea so you need to know the type of expression that you're working with so that was it guys for 16 marks I think uh it was fair enough for you to obtain all these marks that we can see here um after this revision you can answer any type of a question on simplification of algebraic expressions