[Music] [Music] good day everyone and welcome to deaf ed tv i am your teacher joshua i'll be your guide in sharpening your skills and enhancing your minds in order to face the challenges here in grade 8 mathematics your self-learning mojo your pens and your paper with you and let us have a wonderful day of learning last time we talked about factoring polynomials with the greatest common monomial factor today we will factor polynomials that we consider special products but before that i would like you to observe suppose that this square has a side that measures 3 units what do you think is its area the area of the square is just equal to the product of its side multiplied by itself so what is 3 squared or three times three the area of the square is nine square units this illustration shows an example of a perfect square number perfect squares are numbers or expressions that can be expressed to the power of two another example is y raised to six since y raised to six is equal to the square of y cubed by the same logic perfect cube numbers or expressions can be expressed to the power of 3. example is 8 which is equal to 2 raised to the third power now let us try to determine the first five perfect squares and cube whole numbers before proceeding with factoring let us recall how to multiply polynomials what is the product of the sum and difference of two terms the quantity a plus b times the quantity a minus b remember that we can use the foil method to multiply our polynomials we distribute each term to the other group the first term will be the product of the first terms of each binomial a times a which is a squared next is we multiply the outer terms positive a times negative b is negative a b then the inner terms positive b times positive a is positive a b the last term is the product of positive b times negative b which is negative b squared we can simplify the polynomial by combining like terms which will give us a negative a b plus a b equal to zero hence we are left with a squared minus b squared this is a special product that we call the difference of two squares this polynomial is a special product because there is a pattern that is so unique to its form multiplying polynomials is like driving a car forward you are given a starting point and you need to find the product at the last destination while factoring is the reverse process of multiplying polynomials we will drive backwards as we are given with the product and we are asked to find its factors with the previous example we can say that the factors of a difference of two squares a squared minus b squared is the product of the sum and the difference of the positive square roots of each term the quantity of a plus b times the quantity of a minus b look at the square wooden panel the length of each side measures nine inches what is the area of the wooden panel the area is 81 square inches now i want to make a picture frame using this square panel by cutting a square hole inside whose shape and size is equal to the photograph to be placed in since i'm not yet decided on the size of the photograph let us just say its side measures p inches what will be the area of the picture frame in factored form we know that a square hole is to be caught in the square wooden panel so we will subtract the area of the photograph from the area of the wooden panel it is stated that the side of the panel measures 9 inches so the area of that square panel is 81 square inches then we will subtract the area of the photo from the area of the wooden panel the area is p squared what kind of polynomial is this this is a difference of two squares the polynomial can be expressed as the square of nine minus the square of p hence the factored form of the area 81 minus p squared is equal to the quantity of 9 plus p times the quantity 9 minus p but do not forget the unit square inches how about this one right 16 a raised to 6 minus 25 b squared in factored form we start with checking if each term is a perfect square 16 a raised to 6 is equal to the square of a cubed while 25 b squared is equal to the square of 5b the polynomial is a difference of two squares since both terms are perfect squares and the operation is subtraction hence 16 a raised to 6 minus 25 b squared is equal to the quantity of 4a cubed plus 5b times the quantity of 4a cubed minus 5b there are some cases of polynomials that do not seem to be a difference of two squares since the terms are not perfect squares but remember there are other ways to factor your polynomial consider this example factor 3 w squared minus 48 completely are the terms of three w squared minus 48 perfect squares no they are not what else can we do to factor the polynomial note that both terms have a common factor of 3 hence the binomial can be factored using a combination the greatest common monomial factor and the sum and difference of two terms since we have identified that the greatest common factor of 3w squared minus 48 is 3 we can divide the polynomial by the common factor so we can express it as 3 multiplied by the quantity w squared minus 16 then observe that the polynomial factor is a difference of two squares we can further show that the expression three times the quantity of the square of w minus the square of four can you give me the final factored form we can write the polynomial three times the quantity of w plus four times the quantity of w minus four do you know what this is this is a rubik's cube invented in 1974 by erno rubik there are many variations of this puzzle and there are also infinite number of ways to solve this the world record in the fastest solve of a rubik's cube is 4.22 seconds by felix zemdega just like this rubik's cube with different sides or faces the difference of two squares is just one of the several special products the next two involve cubes the difference of two cubes a cubed minus b cubed and the sum of two cubes a cubed plus b cubed but first let us show how to get this special product multiply the quantity of a minus b times the quantity of a squared plus a b plus b squared we can first multiply a to each term of the trinomial a times a squared is a cubed a times a b is positive a squared b and a times b squared is positive a b squared next is to multiply negative b to each term of the trinomial do not forget the signs in multiplying we will have negative b times a squared is negative a squared b then negative b times a b is negative a b squared and negative b times positive b squared is negative b cubed combining like terms we will remove zero pairs positive and negative a squared b and a b squared thus we are left with a cubed minus b cubed we showed that the binomial a minus b multiplied with a trinomial a squared plus a b plus b squared is equal to the difference of two cubes since they are equal we can also say that the reverse is the same meaning that the factored form of the difference of two cubes is given by this pattern the pattern is also similar in factoring the sum of two cubes a cubed plus b cubed what do you notice where do they differ the sign or operation in the binomial factor and the first operation in the trinomial factors are different they both depend on the operation used in the polynomial there is a special trick for us to remember the signs used in this sum or difference of two cubes you can use the mnemonic s o p meaning same opposite positive the binomial factor has the same sign or operation with the given the second sign or the first operation in the trinomial factor is opposite of the binomial the last sign is always positive regardless of the given look at these boxes they are both cubes i would like to put the smaller box inside the bigger box and to keep it safe i will insert some styrofoam chips that will fill the empty space in the bigger box how much styrofoam chips is needed to fill the space and completely secure the smaller box the bigger box has a width of 5 inches but i do not know the size of the smaller box so let us represent it with x since we are asked about the amount of space to be filled we are talking about the volume what is the formula for the volume of a cube the volume of the cube is given by the equation v for volume equal to s cube where s is the length of its side or edge so what are the volumes of the bigger box and the smaller box the volume of the bigger box is equal to the cube of 5 which is 125 cubic inches well the volume of the smaller box is x cubed since we didn't know the exact measure of the side now let us place the smaller box inside the bigger boss what happened to the volume inside the bigger box it got smaller so the amount of space to be filled by styrofoam chips is just the volume of the bigger box minus the volume of the smaller box substituting the values that we have the volume of the empty space is equal to 125 minus x cubed cubic inches we can also write this in factored form since it is a difference of two cubes again we can express 125 as the cube of five minus x cubed we write the binomial factor using the expressions we acquired 5 and x remember that the first sign is the same with the given so we will have 5 minus x then the first term of the trinomial factor is the square of 5 which is 25 the next operation is opposite the binomial what is opposite of minus it is plus then the middle term is the product of 5 and x 5 times x is just 5x the last sign is always positive or a plus sign so finally the last term is the square of x and it is x squared the factored form of the amount of space the styrofoam chips need to fill is the quantity 5 minus x times the quantity 25 plus 5 x plus x squared how about this example factor 40k cubed plus 5. if you notice the operation involved is addition but the terms are not perfect cubed expressions so what else can we do observe that both terms are divisible by five hence a greatest common monomial factor exists we can write the polynomial as five times the quantity eight k cubed plus one then we proceed to factoring the sum of two cubes eight k cubed is the cube of the expression two k well one is equal to the cube of one next what is the binomial factor it is two k plus one since this is a sum of two cubes then the first term of the trinomial is the square of 2k what is it it is 4k squared the next operation must be minus because we need the opposite of the previous one after that we multiply the terms of the binomial 2k times 1 is 2k the final sign is always a positive sign so lastly the third term of the trinomial is one squared and the answer to that is one so we can say that forty k cubed plus one is equal to five times the quantity two k plus one times the quantity of four k squared minus two k plus one and there you have it that's our lesson for today now let us have a recap some polynomials are called special products because they have a certain pattern that we can use in factoring them the difference of two squares a squared minus b squared is equal to the product of the sum and difference of the roots of each term a plus b and a minus b the sum and the difference of two cubes are the product of a binomial and the trinomial factor whose signs depend in the given polynomial remember the mnemonic sop same positive when writing the signs or operations in factoring this special products and now that we are near the end of your lesson prepare your pens and your paper because it is important to evaluate what you have learned i will give you 5 seconds to answer each problem [Music] choose the letter that contains the correct factors of the given polynomial number one factor d squared minus 25 is it a the quantity d plus 5 times the quantity d minus 5 letter b the quantity d plus 25 times the quantity d minus 25 or is it c d plus five times the quantity d squared minus five d plus 25 or d the quantity d minus five times the quantity d squared plus 5 d plus 25 [Music] the correct answer is a the quantity d plus 5 times the quantity d minus 5 next number factor 25 e squared minus 16 is it a e plus 4 times e minus 4 b the product of 5 e plus 4 and 5 e minus 4 c the quantity of the sum 5 e plus 4 times the trinomial 25 e squared minus 20 e plus 16 or d 5 e minus 4 times the trinomial 25 e squared plus 20d plus 16. [Music] the correct answer is b five e plus four multiplied by the quantity five e minus four number three factor twenty c squared minus forty five d squared the choices are a the quantity 2c plus 3d times the quantity 2c minus 3d b the quantity 4c plus 90 times the quantity 4c minus 9d letter c 5 times the quantity of 2c plus 3d times 2c minus 3d or d 5 times the quantity of 4c plus 9d times the quantity 4c minus 9d [Music] correct answer is letter c 5 times the product of the binomials 2c plus 3d and 2c minus 3d how was the activity did you find it difficult just keep on practicing on the examples and assessment found on your self-learning modules remember that in mathematics practice makes you better as an additional exercise and practice at home try answering the activity on your self-learning module about factoring special products the difference of two squares and the sum or difference of two cubes i hope that you have learned a lot in our episode today note that there are many ways to solve a problem and you must focus on the steps and patterns or the process that you have observed in factoring these polynomials with practice and determination i believe that you can ace any lesson in mathematics for our next episode we will be factoring polynomials with three terms or what we call trinomials [Music] remember math is not only about numbers and operations it is an exercise of our minds for us to be critical logical and responsible thinkers again this is teacher joshua reminding you to keep safe have a nice day and see you again next time [Music] [Music] you