[Music] thank you [Music] thank you [Music] good day everyone and welcome back to depth at TV I am your teacher Joshua and I will be your guide in sharpening your skills and enhancing your minds in order to face the challenges in grade 8 mathematics ready or self-learning module your pants and your paper with you and let us have a wonderful day of learning in our previous episode we talked about factoring polynomials that we consider special products can you still remember them they are the difference of two squares the sum of two cubes and the difference of two cubes each is a special product because they can be written in factored form as a product of unique sets of polynomials for today we will Factor another type of polynomial with three terms or a trinomial [Music] know this game this is chess it is played by two players where the objective is to capture the opponent's king through a Checkmate where there are no more available moves for that piece each player has pieces that consist of eight Pawns two Bishops two knights two Rooks a queen and a king there are a few variations of chess but the most common is the game played on an 8 by 8 square board how many squares are there in this chessboard we can easily compute by multiplying the number of rows by the number of columns 8 times 8 or 8 squared is 64. there are 64 squares in an ordinary chessboard some trinomials are like that they are a result of a square of a binomial let us recall how to square the binomial a plus b this can be written as the product of a plus b and itself there are different ways to show this but one of the simplest way is to multiply the expressions using foil method we multiply the first terms in each binomial a makes a is a squared then the outer terms a times B is positive a b next is the inner terms B times a is also positive a b the product of the last terms B times B is positive B squared we will now combine like terms it is usually the outer and inner products that we add in this example a B plus a b is to a b therefore the square of a plus b is equal to a squared plus 2 a b plus b squared though the process is the same the square of the binomial a minus B is equal to a squared minus 2 a B plus b squared these polynomials are special products that we call perfect square trinomials and we apply these patterns to identify the factors of the perfect square trinomial suppose that there will be another variation of a chessboard that will give a total of N squared plus 16 n plus 64 square tiles how many rows should the chess board have we can determine the number of rows of the new chessboard by expressing the polynomial in its factored form we see if this is a perfect square trinomial first is to determine if the first and last terms are perfect squares the first term is already a perfect square of n and 64 is equivalent to 8 squared next is we get twice the product of the expressions and check if it is equal to the middle term of the polynomial what is 2 times n times 8. it is 16 n and it is the same with the middle term of the polynomial we have shown that N squared plus 16 n plus 64 is a perfect square trinomial now we simply write the polynomial as the square of the binomial n Plus 8. we used a plus sign since the middle term is positive therefore the new chess board should have n plus eight rows and columns let us try this example factor the polynomial 8 x squared minus 24 x y plus 18 y squared it is a trinomial but is it a perfect square trinomial look at 8 x squared and 18 y squared this cannot be shown as Expressions to the power of 2. hence they are not perfect squares so there must be another way to factor the polynomial observe what can you see all terms of the polynomial are divisible by two hence it has a greatest common monomial factor of two so factoring out 2 from the polynomial will get to times the quantity 4x squared minus 12 x y plus 9 y squared let us see if the trinomial can further be factored check if the first and the last terms are perfect squares 4x squared is the square of 2x and 9y squared can be expressed as the square of 3y next if we multiply 2 by these expressions what is the product of 2 times 2x times 3y it is 12 X Y which is equal to the middle term of the trinomial Factor note that the first operation used is subtraction which means the binomial also has subtraction now that we have proven that the trinomial is a perfect square trinomial we can say that the polynomial 8X squared minus 24 x y plus 18y squared can be written as 2 times the square of the binomial 2x minus 3y now let us factor X squared plus 10 X Plus 16. let us check if this is a perfect square trinomial the first term is a perfect square of x well the last term can be expressed as 4 squared next let us check if the middle term is equal to twice the product of the acquired expressions what is 2 times x times four it is 8x but wait don't we need 10x to be the middle term what does this mean it means that x squared plus 10 X plus 16 is not a perfect square trinomial there are instances that the trinomial is not a perfect square trinomial but can still be factored let us look at this we have 16 Pawns in chest right we can place and divide them equally in a number of ways earlier we showed 16 can be expressed as 4 squared it means that 16 can be divided equally in four groups of four pawns each but since our polynomial earlier is not a perfect square there must be other factors that may be a solution to our problem can you divide the 16 Pawns in another way this can be divided into two groups of eight one group for white pieces and another group for the black pieces and we will use this concept to factor General trinomials let us have a short exercise before proceeding in factoring General trinomials enumerate pairs of factors whose product is equal to the given number for example six what two numbers can be multiplied such that the result is six 6 can be equal to the product of 1 and 6. or two and three remember also that a positive number can be a product of two negative numbers so six is also equal to the product of negative one and negative six or negative two and negative three how about this one list down two numbers whose product is equal to negative 15. since a negative number is a product of a positive and the negative number its factors must be of different signs negative 15 can be a product of one and negative 15. negative one and positive 15. positive 3 and negative five or negative 3 and positive 5. I believe that you are now ready for the next task quick trivia the closest counterpart of Chess in Filipino Sports is called game of the generals or salpacan invented by sopronio H pasola Jr in 1970. each player contains 21 pieces based on ranks in the Philippine Army it is a game of logic memory and intense strategy the battlefield is an 8x9 rectangular board where the 21 pieces can be placed anywhere in the first three rows of each side and one of the conditions to win is to capture the flag of your opponent when factoring a trinomial that is not a perfect square trinomial we need to identify the following parts the leading term the middle term and the last term with their corresponding signs let us go back to the previous example x squared plus 10x Plus 16. we already know that this is not a perfect square we will now identify the terms the leading term is positive x squared the middle term is positive 10x and the last term is positive 16. next step is to multiply the first and the last terms what is x squared times 16 it is 16 x squared similar to our exercise earlier find two factors such that their product is equal to 16x squared the catch is their sum must be equal to the middle term we know that 16x squared can be written as a product of 8x and 2x now let us check if these are the correct factors what is the sum of 8X into X 8X plus 2X is 10x it means that we got the correct pair of expressions we can now rewrite the polynomial starting with the leading term x squared then the acquired factors plus 8x plus 2X since it is equal to 10x and the last term Plus 16. then we will group them in pairs and get the common factor in each factor X squared plus eight X the common factor is X then multiplied with the binomial X plus eight how about for 2x Plus 16. it can be expressed as 2 times the quantity X plus eight what did you notice both of them has a binomial Factor X plus eight so we can finally Factor it completely as the product of the quantity X Plus 8 times the quantity X plus two I know that the solution may seem long and difficult and there are also different cases of trinomials based on their terms and operations used that is why we need more practice so that the mental and mechanical skills will be instilled to us let us try this example 2x squared plus 5x minus 3. it's clear based on the first term alone that this is not a perfect square trinomial so we can Factor this using the terms in the signs of the trinomial what are the leading middle and last terms do not forget to include the signs the leading term is 2x squared the middle term is positive 5X and the last term is negative three next is we multiply the leading and the last terms what is 2x squared times negative 3. positive 2 x squared times negative 3 is negative 6 x squared now we must look for factors of this expression such that the sum is equal to the middle term positive 5X what could the factors be negative 6 x squared is the product of positive 2X and negative 3x but what is their sum 2X plus negative 3x is negative X which is not equal to 5X so we need to look for other factors of negative 6X squared how about positive X and negative 6x they are definitely factors of 6X squared let us get their sum to check what is X plus negative 6x the answer is negative 5x and it is negative because 6X is larger than x but wait we need positive 5X hence the sign must be wrong the correct Factor should be negative X and positive 6x again find their sum Negative X plus six x is finally positive 5X now we can write the given polynomial as 2x squared minus X plus six x minus 3. we Factor them by grouping what are the common factors in each group the factors are X and three respectively this polynomial is now equal to x times the quantity 2x minus 1 plus 3 times the quantity 2x minus 1. can you now determine the final factored form of 2x squared plus five x minus three the polynomial can be written as the product of the quantity 2x minus 1 times the quantity X plus three that's it for today we are finished in factoring trinomials now let us have a recap some trinomials are special called the perfect square trinomial and to factor a perfect square trinomial first we need to check the first and the last term if they are perfect squares or can be written as a square of a term then twice the product of those terms should be equal to the middle term the factors of a perfect square trinomial is the square of a binomial to factor a general trinomial identify the leading the middle and the last terms then multiply the leading and the last term find factors of the acquired product the sum of the factors should be equal to the middle term if not try another set of factors when the correct factors are identified rewrite the polynomial using the first and the last term and the factors acquired finally Factor by grouping until the polynomial is of the form of a product of two binomials 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 five seconds to answer each question and choose the letter of the correct answer question number one which of the following is a perfect square trinomial is it a x squared minus two X plus four b or x squared plus 12x plus nine c x squared minus 4y plus two y squared or D 9x squared minus 6 x y minus y squared the correct answer is letter B 4x squared plus 12x Plus 9. number two what is the complete factored form of x squared minus 6 x y plus nine y squared is it a the square of the binomial X Plus 2y letter B the product of the binomials X Plus 3y and x minus 3y letter C the square of the binomial x minus 3y for letter d the quantity of x minus 2y times the quantity of X Plus 2y the correct answer is letter C the square of the binomial x minus 3y how was the activity did you find it difficult just keep on practicing on the examples and assessment found on yourself learning modules remember that in math practice makes you better as an additional exercise review the previous lessons that we had and practice on the activities on your self-learning modules because we will apply this Knowledge and Skills in answering word problems involving factoring 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 patterns you have observed the process in factoring these polynomials with practice and determination I believe that you can Ace any lesson in mathematics for our next episode we will have solving problems involving factors of polynomials remember math is not only about numbers and operations it is an exercise for 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 foreign [Music] [Music]