[Music] in this video we will write a polynomial function in standard form also how are we going to factor it okay so i have here p of x is equal to 2x cubed plus 5x squared plus 7x minus five now the terms of a polynomial may be written in any order so however if they are written in decreasing powers of x then the polynomial function is in standard form and polynomial it can be written in any order so halimbawan and situ x 7x but if it's written in decreasing or descending powers of x ibig sabine like this three two one and then zero okay so that is in standard form okay so is nothing exponents now decreasing or descending powers of x then the polynomial is in standard form let's try but before that let us first recall that uh concepts of special products so i know you have already encountered this during your grade seven so grade seven palang alam kona which is the factoring okay so these are the concepts of special products so familiar in highest square of a binomial product of sum and difference of two terms square of trinomial product of binomials product of binomial and trinomials so these are the types of special products so tinaboxan special products [Music] [Music] you know what to do and how to factor polynomials so these are the concepts in factoring polynomials so actually uh factoring polynomials is when you are rewriting a polynomial as a product of polynomials of smaller degree so hapagnag right on polynomial polynomial into smaller degrees okay and then so these are the concepts we have factoring out the greatest common factor factoring by gcf and then difference of two squares sum of two cubes difference of two cubes factoring perfect square trinomials and factoring general trinomials so these are the concepts on how to factor polynomials factoring polynomials at in first quarter in grade 10 and also it's a grade 8 okay so let's have an example so write the given polynomial functions in standard form so when we say standard form your exponents must be arranged in decreasing or descending order so simulation as highest degree up to the least degree so let's have number one i have here y is equal to four x cubed minus 16 x minus four plus x raised to four minus x squared okay so when i want eating then which is the highest degree what is the highest degree so we have here four so the path so that is x raised to four followed by four x raised to three and then four three two at the demand so minus x squared coefficient so minus x squared and then followed by uh first degree we have minus 16x and then lagi pongolina then constant minus four can say this is zero degree okay so x raised to four plus four x cubed minus x squared minus sixteen x minus four so this is the standard form of the given polynomial function let's have number two y is equal to x squared plus 25 plus 10x okay so obviously the highest degree is x squared okay x squared followed by 10 x plus 25 next i have here y is equal to 1 over 6 x raised to 4 minus x squared plus 5 x raised to 5 plus 7 x cubed minus 5. so uh how are you going to arrange this into standard form so being i am here is a fraction but when we are writing a polynomial function in standard form just focus on the exponents okay it's exponents so regardless okay so with let's have first 5x raise to 5 followed by uh 1 over 6 x raised to 4 followed by 7 x cubed minus x squared minus five let's have this one y is equal to the product of x minus three and x plus one squared or the square of x plus one okay so how are you going to arrange this in standard form so um hindi musha pueden go in x plus 1 squared and then followed by x minus 3. i say so you have to expand this first atom square of x plus one so that will become x minus three x plus one x plus one so e big sub negative one now so after that you're going to multiply x plus one x plus one and that is so much multiplying okay so this will become x minus three simplifying nothing x squared plus two x because they combine similar terms plus one now so you have already x minus three x squared plus two x plus one i know next step not then of course you have to multiply so hindi arrangement x squared and then two x and then indi multiply mu we need to eliminate a tongue adding a parenthesis so hunger meron shahib sabin you still need to multiply okay so therefore we will get now how are we going to multiply so one by one x times x squared that is x cubed x times two x that is two x squared x times one that is x and then detonator negative three times x squared that is negative three x squared negative three times two x that is negative six x negative three times one that is negative three now so we will have arranged an atom i so this is one and then negative six so that is negative five okay so but since [Music] hey so just copy x cube and then 2 x squared m is an x squared so hindi mu pointing it again 2x squared and then 3x squared in the populating second degree we can combine these two so positive 2 minus 3 that is negative 1 so we will now have negative x squared and then next i have here also um say uh same literal coefficient so x excellent so combining that in it also x this is one minus six so that that is negative five so negative five x and then just copy your constant minus three now so you um always check your um arrangement if it's written in standard form so x cubed minus x squared minus five x minus three so again exponent and and then multiply tapas and multi characters okay i have here three x plus five and then x plus one so first step multiply by the net and three x times x three x times one and then five times x five times one so you will got um three x squared plus three x plus five x plus five so you will have three x squared so combinating ito this is eight x okay so this is eight x plus five okay so three x squared plus eight x plus five next i have here y is equal to two x times x plus eight so a paganito distribute language so distribute lampuna ten so two x times x and then two x times eight so you will have two x squared plus 16 x because they multiply nothing ito and then x ether next x squared plus x raised to four minus three x raised to six plus x raised to five plus ten so you will have arranged language so first is negative 3x raised to 6 followed by x raised to 5 plus x raised to 4 plus x raised to 2 plus 10. okay so write the following polynomial functions in factored form so this time we will be writing the given polynomial functions in factoid form so if i have here 4x squared minus 81 so this is an example of difference of two squares so how about difference of two squares rewrite natin sha as a square okay so e squared nothing it all a squared not in yen so and on square root and four that is two so i x and then square root of 81 that is nine okay now so this will become your um hey this will become your first term and this is your last term okay square root last term so you will have two x plus nine and then two x minus nine okay so as you can see para hoshan is a first second term and then difference of two squares it's always plus and minus so therefore our um our factored form is 2x plus 9 2x minus 9. okay so this is naman an example of the sum of two cubes so e big sub n cube e big sub b and perfect cube it see 2 by itself 3 times so 2 times 2 4 times 2 8 okay so and then you cube root me b cube that is b okay and then cube root of 27 that is 3 and then cube cube is c so we now have our two b and three c so anan gaga be not in gen so this will serve as our x and y so play like nothing young missing terms you know so young x epic sub in this is 2b this is 3c so so supply not n so this is x so this is 2b this is y so this is 3c this is excellent so this is 2b squared and then this is 2b this is 3c this is 3c now so since many times square d2 we still need to simplify this because okay so we will have 2b plus 3c and then 2 squared is 4 and then b squared minus times nothing it is a multiplication 2 times 3 that is 6 and then bc and then ito simplify did not end this is 3 squared so 9 3 times 3 is 9 and then c squared okay so therefore the factored form is two b plus three c and four b squared minus six b c plus nine c squared okay let's have this one so this is demand and example of perfect square trinomial [Music] what is the square root of 16x squared that is 4x what is the square root of 81 that is 9 and then ito happy molang todito 4x and 9 tapasya times nathan check nathan dapat max 72x so 2 times 4 8 8 x times 9 that is 72 x so therefore this is a perfect square to the trinomial kapaging results a middle term hindi push a perfect square trinomial so you will be having general training so if a factor form okay so now and it's a perfect square trinomial then you can have just i know just follow this pattern x y so four x and nine squared okay perfect square trinomial perfect square trinomial so this is what is the square root of 25 m squared that is 5 m okay and then what is the square root of 4n squared that is 2n okay it should be 2n so it's 2n sorry for the typo okay so it allegating so it should be all right so we have nothing laser okay so we have i checked in 2 times 5 10 times 2 that is 20 and then minus okay so since uh satisfied shadows are adding middle and then squared okay so this is the pattern let's have the last example so i have here y is equal to x raised to four minus four x squared minus forty five so um for factored form since we can rewrite this as x squared raised to two bachelor two times two this is still x raised to four okay why do we need to rewrite this okay so we will let x squared is equal to x x squared okay we minus four that plus x squared oled there gonna be nothing x would get given one nothing x minus forty five now so hapag final or nothing it oh we will have x minus nine x plus let x squared is equal to x so human x naught and papadi not in x x squared pinata x squared so we now have x squared minus nine x squared plus five okay so our factored form is x plus three x minus 3 and x squared plus 5. so why do we have x plus 3 x minus 3. so this is an example of difference of 2 squared so what is the square root of x squared x what is the square root of nine three and then laggy pushing plus and minus so exponential and then just copy this one away this is not a perfect square so hindi push a difference of two squares so it is so therefore our final answer is x plus three x minus three x squared plus five thank you for watching this video i hope you learned something don't forget to like subscribe and hit the notification bell for updated ko for more video tutorials this is your guide in learning your math lessons your walmart channel