Polynomials. Polynomials. Curriculum Ready ACMNA:

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1 Polynomils Polynomils Curriulum Redy ACMNA: 66

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3 Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression is non-negtive integer. Answer these questions, efore working through the hpter. I used to think: Wht is the degree nd leding oeffiient of polynomil? If n expression is divided y ftor, then wht is the reminder? How do the ftors of polynomil relte to its solution in n eqution? Answer these questions, fter working through the hpter. But now I think: Wht is the degree nd leding oeffiient of polynomil? If n expression is divided y ftor, then wht is the reminder? How do the ftors of polynomil relte to its solution in n eqution? Wht do I know now tht I didn t know efore? 100% Polynomils Mthletis 100% 3P Lerning K 1 1 SERIES TOPIC

4 Polynomils Bsis Wht is Polynomil? It is lredy known tht inomil ontins two terms (eg. x + ) nd trinomil ontins three terms (eg. x + x + ). Now, polynomil n hve ny numer of terms, ut there re few rules. Here is generl polynomil: The highest power is lled the degree of the polynomil The powers must e non-negtive intergers. no frtions, no negtive numers A polynomil n only hve one vrile (x) Px () x n n n-1 n- = + n -1x + n -x x + x+ 1 0 The oeffiient of the highest power (n ) is lled the leding oeffiene 1 to n re the oeffiients of the polynomil. The oeffiients n y ny rel numer This is onstnt term If the leding oeffiient (n) is 1, then the polynomil is lled moni. The P(x) mens the polynomil is lled P nd is in terms of the vrile x. Here re some exmples. Answer the following questions out P(x) nd Q(x) elow: d 4 6 Px () = x - 5x + 3x - x+ 7 Qx () x nd = - x + 4x -8x - 3x + 6x- Wht is the degree of eh polynomil? The degree (highest power) of P(x) is 4. The degree (highest power) of Q(x) is 6. Wht is the leding oeffiient of eh polynomil? The leding oeffiient of P(x) is. The leding oeffiient of Q(x) is 1. Whih polynomil is moni? Q(x) is moni euse it hs leding oeffiient of 1. Wht is the onstnt term of eh polynomil? The onstnt term of P(x) is 7. The onstnt term of Q(x) is -. In the nottion of polynomils, the vrile is written inside rkets. Just like P(x). If nything else is written inside the rket, sustitute this vlue into eh vrile. Here is n exmple. Find the following vlues if P^xh= x - 4x + 3x + 5 P( ) P( - 1) P^h = ^h - 4^h + 3^h+ 5 = = 11 P^- 1h= ( -1) -4(- 1) + 3^- 1h+ 5 = =-4 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

5 Polynomils Questions Bsis 1. Explin why the following re not polynomils. 4x - + x+ 1 3x 3 - x - 1 x 3-7x -5 d x + y+ 3 e 3 x + x + 6 f 1 4x x x 100% Polynomils Mthletis 100% 3P Lerning K 3 1 SERIES TOPIC

6 Polynomils Questions Bsis 5 4. Anwser the following questions out P^xh =- 3x + 8x - x + 4x+ 6 nd R^th = t - t + t - 5. Wht is the onstnt term in eh polynomil? Wht is the leding oeffiient in eh polynomil? Are either of the polynomils moni? d Wht is the degree of eh polynomil? e Find P( - 1). f Find P( ). g Find R (1). h Find R (3). 4 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

7 Polynomils Bsis Adding nd Sutrting Polynomils? When dding nd sutrting polynomils, simply ollet like terms. Let P^xh= x 5 + x 4-3x 3 + 7x -6x- 1 nd Q^xh=- x 6 + x 5 + 7x 4 + 5x x - 8x + 4 Find P^xh+ Qx ^ h Px ^ h+ Qx ^ h= ^x + x - 3x + 7x -6x- 1h+ ^- x + x + 7x + 5x + 10x - 8x + 4h = x + x - 3x + 7x -6x-1- x + x + 7x + 5x + 10x - 8x =- x + ^x + x h+ ^x + 7x h+- ^ 3x + 5x h+ ^7x + 10x h+ ^-6x- 8xh+- ^ 1+ 4h =- x + 3x + 9x + x + 17x - 14x + 3 Find P^xh- Qx ^ h Px ^ h- Qx ^ h= ^x + x - 3x + 7x -6x-1h-^- x + x + 7x + 5x + 10x - 8x + 4h = x + x - 3x + 7x -6x- 1+ x -x -7x -5x - 10x + 8x = x + ^x - x h+ ^x - 7x h+ ^-3x - 5x h+ ^7x - 10x h+- ^ 6x+ 8xh+ ^-1-4h = x -x -5x -8x - 3x + x- 5 Multiplying Polynomils Use the distriutive lw to multiply polynomils. Let P^xh=- 3x+ 4 nd Q^xh= x - 3x + 1. Find P^xh# Qx ^ h. Px ^ h# Qx ^ h= ^- 3x+ 4h^x - 3x + 1h =-3x^x - 3x+ 1h+ 4 ^ x - 3x + 1h = ^- 6x + 9x - 3xh+ ^8x - 1x + 4h =- 6x + ^9x + 8x h +- ^ 3x- 1xh + 4 Find Q^xh# Px ^ h. =- 6x + 17x - 15x + 4 Qx ^ h# Px ^ h= ^x - 3x+ 1h^- 3x + 4h = x ^- 3x+ 4h-3x^- 3x+ 4h+ 1^- 3x + 4h = ^- 6x + 8x h+ ^9x - 1xh+ ^- 3x + 4h =- 6x + ^8x + 9x h +- ^ 1x- 3xh + 4 =- 6x + 17x - 15x + 4 So Px ()# Qx () = Qx ()# Px (). This is lwys true! Sometimes Px ()# Qx () is written s Px (): Qx () 100% Polynomils Mthletis 100% 3P Lerning K 5 1 SERIES TOPIC

8 Polynomils Questions Bsis Simplify ^x + 4x - x + 5x+ 9h+ ^x - 3x + h. 4. Let T^xh= x 5 + 3x 4 - x 3 + 3x - x+ 3 nd Q^xh = 3x 5 - x 4 + 6x 3 + 7x - 4x+ 5. Find T^xh+ Qx ^ h. Let Sx ^ h= Tx ^ h+ Qx ^ h. Find S^-1h. Find T^- 1h+ Q^-1h. Wht do you notie? 6 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

9 Polynomils Questions Bsis 5. Use Tx ( ) nd Q^xh in the previous question to nswer these questions: Let D^xh= Tx ^ h-qx ^ h. Find D^h. Does D^h= T^h -Q( )? Wht is the degree, leding oeffiient nd onstnt term of D^xh? 100% Polynomils Mthletis 100% 3P Lerning K 7 1 SERIES TOPIC

10 Polynomils Questions Bsis 6. Let P^xh= x + 1 nd Q^xh = 3x -x - 5. Find P^xh: Qx ^ h. Find P^h: Q^h. Wht is the degree, leding oeffiient nd onstnt term of P^xh: Qx ^ h? 8 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

11 Polynomils Questions Bsis 3 7. Let M^yh= 3y- nd N^yh = 4y - y + 3. Find M^-3h nd N^-3h. Find M^-3 h# N( -3). Let Ay ^ h= My ^ h# Ny (). Find Ay (). d Find A( - 3). Is this the sme s M( -3)# N( - 3)? e Wht is the degree, leding oeffiient nd onstnt term of Ay ^ h? 100% Polynomils Mthletis 100% 3P Lerning K 9 1 SERIES TOPIC

12 Polynomils Knowing More Dividing Polynomils Polynomil division is sed on long division. Using long division to find 13' 5: Quotient g Divisor It is esy to see tht 13 = 5^4h + 3. This is lwys true: Dividend = Divisor # Quotient + Reminder. Dividend Reminder Here is n exmple using polynomils: Rewrite the dividend in terms of the divisor, quotient nd reminder. Quotient Divisor x - x + 1 r 6 x+ 4 x + x - 7x+ 10 g Reminder Dividend so x + x - 7x+ 10 = ( x+ 4)( x - x+ 1) + 6 The question is: How is this quotient found? It is found with long division using the sme proess s the numers ove: Find this quotient using long division: Step 1: Divide x 3 (first term of dividend) y x (first term of the divisor). This gives the first term of the quotient. ( x + x - 7x+ 10) ' ( x+ 4) Step : Multiply x y x + 4 (divisor) nd sutrt it from the dividend Step 3: ring the - 7x (the next term) down Dividend Quotient Divisor Step 4: Divide x - (first term of new dividend) y x (first term of the divisor) nd write this in the quotient. Step 5: Multiply x + 4 (divisor) y - x (the new quotient term) nd sutrt from -x - 7x (the new dividend) Step 6: ring the + 10 (the next term) down Step 7: Divide x (first term of new dividend) y x (first term of divisor) nd write this in the quotient Step 8: Multiply 1 y x + 4 (divisor) nd sutrt from x + 10 (new dividend) Step 9: Stop when the differene is onstnt (6 in this exmple) x - x+ 1 x+ 4 x + x - 7x+ 10 g x + 4x -x -7x -x -8x x + 10 x Reminder ` x + x - 7x+ 10 = ( x+ 4)( x - x+ 1) + 6 Dividend = Divisor # Quotient + Reminder 10 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

13 Polynomils Knowing More After eh step the expression under the dividend eomes the new dividend. The divisor never hnges. Here is nother exmple: Find ^x 3 + 3x-3x-h' ( x- 3) 3 Step 1: Divide x y x. This gives the first term of the quotient. Step : Multiply x y x- 3 (divisor) nd sutrt it from the dividend Step 3: ring the - 3x (the next term) down Step 4: Divide 6x (first term of new dividend) y x (from divisor) nd write in the quotient. Step 5: Multiply x + 3 y 3x (new quotient) nd sutrt from 6x - 3x (new dividend) Step 6: ring the - (the next term) down x + 3x + 3 x- x + 3x -3x- g x - 3x 6x - 3x 6x - 9x 6x - 6x So x + 3x -3x- = (x- 3)( x + 3x+ 3) + 7 Step 7: Divide 6x (first term of new dividend) y x (first term of divisor) nd write this in the quotientt Step 8: Multiply 3 y x - 3 (divisor) nd sutrt from 6x - (new dividend) Step 9: Stop when the differene is onstnt (7 in this exmple) Now this eomes prtiulrly interesting when the reminder is zero. If the reminder is zero, then the divisor is tully ftor of the dividend. This is euse if the reminder is zero then there is tully no reminder, whih mens the divisor would hve to e ftor. Is ^x + 3h or ^x - 3h ftor of x + x -10x - 6? Divide x + x -10x - 6 y x + 3 nd then y x - 3 nd see whih quotient hs reminder of zero: x -x - 1 x+ 3 x + x -10x-6 g x + 3x -x -10x -x - 9x -x -6 -x -3-3 x + 4x + x- 3 x + x -10x-6 g x + 3x 4x - 10x 4x - 1x x - 6 x So x + x -10x- 6 = ( x+ 3)( x -x-1) -3 So x + x -10x- 6 = ( x- 3)( x + 4x+ ) There is reminder of -3 There is no reminder ` x + 3 is not ftor ` x - 3 is ftor 100% Polynomils Mthletis 100% 3P Lerning K 11 1 SERIES TOPIC

14 Polynomils Questions Knowing More 1. Use the following to nswer the questions elow: 3x + x + 5 r 11 x-1 3x - x + 4x+ 6 g Wht is the divisor? Wht is the reminder? Wht is the dividend? d Wht is the quotient?. Find these quotients (rewrite s Dividend = Divisor # Quotient + Reminder). ( x - 3x + 4x-5) ' ( x- ) (x + 5x + x- 1) ' ( x+ 4) 1 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

15 Polynomils Questions Knowing More 3. Find the following quotient: 4 x - x + 5x - 6x+ 8 x - 1 Rememer = ' 4 Complete the following: x - x + 5x - 6x+ 8 = ( x - 1)( ) + 100% Polynomils Mthletis 100% 3P Lerning K 13 1 SERIES TOPIC

16 Polynomils Questions Knowing More 4. How do you know if divisor is ftor? 5. Show tht x + 7 is ftor of x + 17x + 0x Show tht x - 4 is ftor of 6x - 8x + 4x K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

17 Polynomils Using Our Knowledge The Reminder Theorem Here is some stndrd mthemtil nottion: Let Px ^ h= dividend Qx () + Let Qx ^ h= quotient x- g Px () Let x- = divisor, where is ny onstnt Let = reminder. The reminder will lwys e onstnt if the divisor is liner. Now So this mens nd for P^h Dividend = Divisor # Quotient + Reminder Px ^ h = ^x- hqx ^ h + P ^ h = ^- hq ^ h + = 0. Q ^ h + = So P^h is the reminder of Px ^ h ' ( x- ). This mens the reminder n e found without doing long division. Here re some exmples Find the reminder of the following quotient ^x + x - 3x+ 8h' ( x-) Let Px ^ h= x + x - 3x + 8 Thedivisor is x-. ` =. P^h= + ^ h - 3 ^ h + 8 = 18 ` ^x + x - 3x+ 8h ' ( x-) hs reminder of 18. Find the reminder of the following quotient ^x + x - 3x+ 8h' ( x+ ) Let Px ^ h= x 3 + x - 3x + 8 The divisor is x+, or x-^- h. ` =-. = 14 ` ^x + x - 3x+ 8h ' ( x+ ) hs reminder of 14. P^- h= (- ) + ( -) -3^- h % Polynomils Mthletis 100% 3P Lerning K 15 1 SERIES TOPIC

18 Polynomils Questions Using Our Knowledge 1. Find the reminder of ^x + 3x -6x-8h ' d^xh if: dx ^ h= x -3. dx ^ h= x + 1. dx ^ h= x +. d dx ^ h= x -. e dx ^ h= x -7. f dx ^ h= x + 4. g Whih divisors ove re ftors? (leve reminder of 0) 16 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

19 Polynomils Using Our Knowledge The Ftor Theorem Rememer it is ftors we re interested in, nd ftors use reminder to e zero. The reminder theorem sys tht the vlue of P^h is the reminder of Px ^ h ' ( x- ). So if P ^ h = 0 then the reminder of Px ()'( x- ) is 0. This mens ^x- h would e ftor. The Ftor theorem sttes: ^x- h is ftor of P^xh if nd only if P ^ h = 0. Here is n exmple: Find ftor of x + 3x -6x -8 Let Px () = x + 3x -6x -8 =-10 ` P() 1! 0 ` P() 1 = () -61 ()-8 ` ` P( - 1) = (- 1) + 3( -1) -6(-1)-8 = 0 P( - 1) = 0 ` x - 1 is not ftor ` x-- ^ 1h= x+ 1is ftor Use tril nd error to find the vlue for so tht P ^ h = 0. Try = 1, then =- 1, then =, then =- nd so on. Find ftor for x + 4x -11x-30 nd perform long division to ftorise it into three rkets with no reminder Let Px ^ h= x + 4x -11x -30 P^1 h =-36. So x - 1 is not ftor of P^xh P^- 1h =-16. So x-- ^ 1h = x+ 1 is not ftor of P^xh P^ h =-8. So x - is not ftor P^- h = 0. So x-- ^ h = x+ is ftor x + x - 15 x+ g x + 4x -11x-30 x 3 + x x - 11x x + 4x -15x x ` ` Px () = ( x+ )( x + x -15) Px () = ( x+ )( x- 3)( x + 5) This n e ftorised like n ordinry qudrti trinomil 100% Polynomils Mthletis 100% 3P Lerning K 17 1 SERIES TOPIC

20 Polynomils Questions Using Our Knowledge. Answer the following questions out P^xh = x -x - 9x Find the vlue of: P() 1 P( -1) Is ^x - 1h or ^x + 1h ftor of P^xh? Use long division with the ftor ove to ftorise P^xh into three rkets. 18 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

21 Polynomils Questions Using Our Knowledge 3. Find liner ftor for these polynomils: 3x + 8x - 33x + 10 x + 19x + 3x - 1 x -3x -65x % Polynomils Mthletis 100% 3P Lerning K 19 1 SERIES TOPIC

22 Polynomils Questions Using Our Knowledge 4. Ftorise the following into three rkets with no reminder. 3x - 19x + 16x + 0 4x -5x - 47x K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

23 Polynomils Thinking More Ftorising Polynomils Dividing polynomils, the ftor theorem nd the reminder theorem re used to ftorise polynomils into rkets. This n e done for polynomil with ny degree (highest power). Here is n exmple with degree 4 polynomil. Ftorise this polynomil into liner ftor rkets: P^xh= x 4 + 3x 3-15x - 19x+ 30 P^1 h = 0 nd so ^x - 1h is ftor of P^xh. x + 4x -11x x- 1g x + 3x -15x - 19x x - x 4x - 15x 4x - 4x -11x -19x - 11x + 11x - 30x x + 30 There is no reminder, s we expeted 0 So ording to P^x h = Divisor # Quotient + Reminder Px () = ( x- 1)( x + 4x -11x- 30) Needs to e ftorised gin Let Ax ^ h= x + 4x -11x -30 A^- h = 0 nd so ^x + h is ftor of Ax ^ h x + x - 15 x+ g x + 4x -11x-30 x 3 + x x - 11x x + 4x -15x x -30 There is no reminder, s we expeted 0 So Ax ^ h = ^x+ h^x + x -15h This mens P^xh = ^x- 1h^x+ h^x + x- 15h Ftorise s qudrti Px ^ h = ^x- 1h^x+ h^x- 3h^x+ 5h Whih hs een ftorised into 4 rkets of liner ftors. 100% Polynomils Mthletis 100% 3P Lerning K 1 1 SERIES TOPIC

24 Polynomils Thinking More Polynomil Equtions The whole point of ftorising polynomils is to solve polynomil equtions. Here is n exmple. Solve the following eqution 4 x + 3x -15x - 19x+ 30 = 0 From the previous exmple the polynomil n e ftorised to eome: ^x- 1h^x+ h^x- 3h^x+ 5h= 0 ` x - 1 = 0 x + = 0 x - 3 = 0 x + 5 = 0 or or or x = 1 x =- x = 3 x =-5 When ftorising using long division, e reful not to leve out power. Look t this exmple: Solve this eqution Let Px ^ h= x - 1x+ 0 3 P^1 h = 0 so x-1 is ftor of P^xh. x 3-1x+ 0 = 0 Notie, there is no x term inp^xh. No power n e left out of long division, so rewrite Px ^ h = x + 0x - 1x+ 0. x 3-1x+ 0 = 0 ` ( x- 1) ^x + x - 0h= 0 x + x - 0 x- 1g x + 0x - 1x+ 0 x - x x x - 1x - x - 0x + 0-0x There is no reminder, s we expeted Ftorise s qudrti ` ( x-1) ^x- 4h^x + 5h= 0 ` x = 1 or x = 4 or x =-5 When solving polynomil equtions, follow these three steps. Step 1. Mke sure the right hnd side is 0 Step. Rewrite the polynomil to inlude missing powers Step 3. Ftorise. K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

25 Polynomils Questions Thinking More 1. Let P^xh= x -5x - 18x + 7. Find numer so tht P ^ h = 0. Bsed on, find liner ftor of P^xh. Ftorise P^xh. d Solve the eqution x -5x - 18x + 7 = % Polynomils Mthletis 100% 3P Lerning K 3 1 SERIES TOPIC

26 Polynomils Questions Thinking More 4. Let P^xh = x + x -13x - 14x+ 4. Given tht P^- h = 0, ftorise P^xh into four liner ftors. 4 Solve the eqution x + x -13x - 14x+ 4 = 0. 4 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

27 Polynomils Questions Thinking More 3. Solve x -x - 15x + 36 = 0 y ftorising. 100% Polynomils Mthletis 100% 3P Lerning K 5 1 SERIES TOPIC

28 Polynomils Questions Thinking More 4. Solve x + 3x - 4 = 0 y ftorising. (Be reful of the missing power) 6 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

29 Polynomils Questions Thinking More 5. Solve x + 8x -10x- 40 = x + 4x + 8 y ftorising. Hint: move everything to one side nd equl to 0 first. 100% Polynomils Mthletis 100% 3P Lerning K 7 1 SERIES TOPIC

30 Polynomils Answers Bsis: 1. The first term hs negtive power -, 5. so the expression is not polynomil. d The first term hs frtionl power 3, so the expression is not polynomil. The power in the first term 3 x is not neessrily n integer, so the expression in not polynomil. The polynomil must hve only one vrile. This expression hs x nd y. 6. Bsis: D^h=-30 Yes, D^h= T^h-Q^h The degree of D(x) is 5 Leding oeffiient is -1 Constnt term is - 3x -x -x -x e f The expression hs frtionl index so it is not polynomil. The first term of the expression hs negtive power so it is not polynomil. 7. The Degree is 4 The leding oeffiient is 3 The onstnt term is 5. M( - 3) =-11. Px ^ h: 6 R^th: -5 N^- 3h=-10 Px ^ h: -3 R^th: 1 11 d Px ^ h : No, the leding oeffiient is not 1 R^th : Yes, the leding oeffiient is 1 Px ^ h :5, the vlue of the highest power R^th :4, the vlue of the highest power d 4 1y -8y - 3y + 11y-6 A^- 3h= 11 A(-3) is the sme s M^-3h# N^-3h e g P( - 1) =- 5 f P( ) =-6 R() 1 =- 3 h R( 3) = 67 e The degree is 4 The leding oeffiient is 1 The onstnt term is x + 5x -x - 3x + 5x + 11 Knowing More: x + x + 4x + 10x - 6x + 8 S^- 1h= 16 T^- 1h+ Q^- 1h= 16 Notie T^- 1h+ Q^- 1h= S^-1h 1. d x - 1 is the divisor 11 is the reminder 3x - x + 4x + 6 is the dividend 3x + x+ 5 is the quotient 8 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

31 Polynomils Answers Knowing More: Using Our Knowledge:. ^x - 3x + 4x-5h ' ^x-h 3. x - = ^x-h # ^x - x + h -1 x + 3 ^x + 5x + x- 1h ' ^x+ 4h x + 4 = ^x+ 4h^x - 3x + 13h Px ^ h = ^x- h^3x+ h^x -5h 3. 4 x - x + 5x - 6x+ 8 Px ^ h = ^x+ 3h^4x-1h^ x -4h = ^x-1h^x - x + 4x- h + 6 Thinking More: 4. A divisor is ftor if the reminder is zero. 1. P( 3) = 0 5. x + 7 is ftor of x + 17x + 0x - 7 euse when x + 17x + 0x - 7 is divided y x + 7, the reminder is zero. d x - 3 is liner ftor. Px () = ^x-3h^x- 6h^x + 4h x = 3, x = 6 nd x =-4 6. x - 4 is ftor of 6x - 8x + 4x - 4 euse when 6x - 8x + 4x - 4 is divided y x - 4, the reminder is zero.. Px () = ^x+ h^x- 1h^x+ 4h^x- 3h x = 1 or x = 3 or x =- or x =-4 1. Using Our Knowledge: x = 3 or x = 3 or x =-4 8 d 0 4. x = 1 or x =- or x =- e 440 g ^x + 1h, ^x + 4h nd ^x - h re ftors. f 0 5. x =- or x =- 8 or x = 3. P() 1 = 0 P( - 1) = 56 x - 1 is ftor of P^xh sine P() 1 = 0 ^x-1h^x- 6h^x + 5h 100% Polynomils Mthletis 100% 3P Lerning K 9 1 SERIES TOPIC

32 Polynomils Notes 30 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

33 Polynomils Notes 100% Polynomils Mthletis 100% 3P Lerning K 31 1 SERIES TOPIC

34 Polynomils Notes 3 K 1 100% Polynomils SERIES TOPIC Mthletis 100% 3P Lerning

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36 Polynomils