Operazioni con i polinomi - Istituto Comprensivo Statale ... · Operazioni con i polinomi - 3...

Operazioni con i polinomi - 1

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http://creativecommons.org/licenses/by-nc-nd/3.0 (Attribution-Noncommercial-No Derivative Works 3.0) La riproduzione di tutto o parte dei contenuti potranno avvenire solo senza alcun scopo di lucro e dovranno riportare l’attribuzione all’autore ed un link a UbiMath e/o a quella dell’autore/i originario.

Operazioni con i polinomi

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1. a + b ∙ a − b − b ∙ a − b = 𝒂𝟐 − 𝒂𝒃

2. a − b ∙ a + 1 − a ∙ a + 1 = −𝒂𝒃 − 𝒃

3. 2x − 2y ∙ 2x − 2y − 2x ∙ x − y = 𝟒𝒙𝟐 − 𝟔𝒙𝒚 + 𝟒𝒚𝟐

4. 3a2 ∙ a + b − 3b ∙ a2 − 2b = 𝟑𝒂𝟑 + 𝟔𝒃𝟐

5. 4x3: 4x2 x − y + 2x ∙ 2xy = 𝟏

6. 2 3 3 2 22 2x x x y y x y x y x y y 𝟒𝒙𝒚

7. 2222222 2925396 xyxyxyyxyxxyyx 𝟒𝒙𝟐𝒚 − 𝟐𝒙𝒚𝟐

8. cbacbacba 426532 −𝟖𝒂 + 𝟐𝒃 + 𝟑𝒄 (*)

9. 1

3a2 ∙

1

2a + b −

1

2b ∙

1

3a2 +

1

2b =

𝟏

𝟔𝒂𝟑 +

𝟏

𝟔𝒂𝟐𝒃 −

𝟏

𝟒𝒃𝟐

10. 1

2a + 2b + 2 ∙ a + b =

𝟓

𝟐𝒂 + 𝟒𝒃

11. − 1

2a + 2b − 2 ∙ a − b = −

𝟓

𝟐𝒂

12. x −2

3y ∙ y −

1

3x −

1

3x ∙ −

2

3y − x =

𝒙𝒚 −𝟐

𝟑𝒚𝟐

13.

abaaab 3

5

2

6

1

2222

2

1

15

44

5

2babaa

14.

2222

2

143122

4

34 xxxxxx

2

15

4

152 2 xx

(*)

15. 6 9 3 5 221

5

7

15

2 2 2 2 2 2 4 2x y xy x y x y xy x y z xy z

4x2y − 2xy2

16. 1

55

2

3

7

15

5

63

2

3

2 3 2 3 2 3 2 3 2 2 2x y x y x y x y xy xy xy

𝟏𝟎

𝟑𝒙𝒚

17. 32175236 xxxx

−𝒙 + 𝟏𝟑 (*)

18.

3432223

10

9

5

6

3

2

3

5

4

3

9

10babaabbaab

𝟏

𝟐𝒂𝟑𝒃 − 𝟐𝒂𝟐𝒃𝟓 (*)

19. 4 23

23

2

3

1

3

1

262 2 2 2 2 2 2x y x xy y x xy xy x y

𝟐𝒙𝟒

mailto:[email protected]

http://creativecommons.org/licenses/by-nc-nd/3.0




20.

yxyxyxyyyx

3

2

2

1

3

2

3

42

3

1

3

1

3

1

2

1

𝟑

𝟐𝒙 + 𝒙𝒚 − 𝒚 −

𝟓

𝟗𝒚𝟐

21.

2222222

5

2

6

1

3

2

2

32

4

3

2

1

5

2baaababbaab

−𝒂𝒃

22.

babaabccba

2

9

3

4

3

4

5

2

15

2

5

4 2223

23. 𝒂 + 𝟐𝒃 ∙ 𝒂 − 𝟐𝒃 ∙ 𝟑𝒃 − 𝟐 − 𝟒𝒃𝟐 ∙ (𝟐 − 𝟑𝒃) =

24.

423633423

2

22 2:22:3

4:

3

2xbaxbaxyyxx

(*)

25. 2325231 22 xxxxxx

26. bababababaa 511254332 2

27.

3423534

3

1:

3

425

3

1xxaaxxax (*)

28.

yxxyyx 2

3

5

4

3

3

2

4

525

5

4

29.

yxyxyx 64

2

13

3

2

(*)

30.

2

2

2222

3

1

2

1

5

3

5

2

2

1

5

1

5

1

4

1

5

2yxxyxyyxyx

yx

25

4

10

1 3 (*)

31.

baaabbababaabaa 23222

3

1

4

32

4

1

3

1

32234

24

35

48

13bababa

(*)

32.

xyyxyxyxxyyx 63

3

132

2

1

3

2 2

3223

3

2

3

17

3

4

2

1yxyyxx

(*)

33. −2

3𝑥𝑦

1

2𝑥3𝑦 − 2𝑥3𝑦 + 𝑥3𝑦 −𝑥𝑦 −

1

2𝑥𝑦 − 2𝑥4𝑦2 = −

𝟓

𝟐𝒙𝟒𝒚𝟐

34. −𝑦2 − 1

2𝑥2 −

3

4𝑥2 − 4𝑦2 + 𝑥𝑦 −

3

2𝑥𝑦 −

2

3𝑥2 ∙

3

5= −

𝟏

𝟒𝒙𝟐 +

𝟑

𝟐𝒙𝒚 − 𝟑𝒚𝟐

35.

15

62

3

12

5

2 xxx −

𝟏

𝟑𝒙 −

𝟏

𝟑𝒙

36.

3

13

2

1222 xxx

𝒙 + 𝟐

37.

xxx

10

7

5

2

2 −

𝟐

𝟓

(*) gentile concessione di stringher.blog.kataweb.it/ - Commissione e-learning IPSSCART B. Stringher – Udine

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http://stringher.blog.kataweb.it/




Soluzioni

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a + b ∙ a − b − b ∙ a − b =

= a2 − b2 − ab + b2 =

= a2 − b2 + b2 − ab =

= a2 − ab

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

a − b ∙ a + 1 − a ∙ a + 1 =

= a2 + a−ab − b − a2 − a =

= −ab − b

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

2x − 2y ∙ 2x − 2y − 2x ∙ x − y =

= 4x2 − 8xy + 4y2 − 2x2 + 2xy =

= 4x2 − 2x2 − 8xy + 2xy + 4y2 =

= 4x2 − 6xy + 4y2 Grazie Manuela B. per la segnalazione 10.1.2008

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3a2 ∙ a + b − 3b ∙ a2 − 2b =

= 3a3 + 3a2b − 3a2b + 6b2 =

= 3a3 + 6b2

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4x3: 4x2 x − y + 2x ∙ 2xy =

= 4x3: 4x3 − 4x2y + 4x2y =

= 4x3: 4x3 = 1

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

2 3 3 2 22 2x x x y y x y x y x y y

2 3 3 3 3 2 2 4

2 3 2 2 3 4

4

2 2 2 2 2 2

2 2 2 2 2 2

x x xy xy y x xy xy y y

x x x xy xy y y y

xy

oppure 2 3 3 2 22 2x x x y y x y x y x y y

= -2x2 +3x

2 +3xy -3xy +3y

2 -x

2 +4xy -4y

2 +y

2 =

= +3x2 -2x

2 -x

2 +3xy -3xy +4xy +3y

2 -4y

2 + y

2 =

= 4xy

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2222222 2925396 xyxyxyyxyxxyyx

22

22

222222

222222

222222

24

2536

929536

925396

925396

xyyx

xyyx

xyxyxyyxyxyx

xyxyyxyxxyyx

xyxyyxyxxyyx

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

cbacbacba 426532

cba

cba

cccbbbaaa

cbacbacba

328

425413462

42543462

44426532

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

1

3a2 ∙

1

2a + b −

1

2b ∙

1

3a2 +

1

2b =

=1

6a3 +

1

3a2b −

1

6a2b −

1

4b2 =

=1

6a3 +

2 − 1

6a2b −

1

4b2 =

=1

6a3 +

1

6a2b −

1

4b2

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

1

2a + 2b + 2 ∙ a + b =

=1

2a + 2b + 2a + 2b =

=1 + 4

2a + 2 + 2 b =

5

2a + 4b

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

− 1

2a + 2b − 2 ∙ a − b =

= −1

2a − 2b − 2a + 2b =

=−1 − 4

2a = −

5

2a

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=






𝑥 +2

3𝑦 ∙ 𝑦 −

1

3𝑥 −

1

3𝑥 ∙ −

2

3𝑦 − 𝑥 =

= 𝑥𝑦 −1

3𝑥2 +

2

3𝑦2 −

2

9𝑥𝑦 +

2

9𝑥𝑦 +

1

3𝑥2 =

= xy +2

3y2

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

abaaab 3

5

2

6

1

2222

2222

22222

22222

2

1

15

44

5

2

2

1

15

451

5

2

2

13

15

1

5

2

35

2

2

1

15

1

babaa

babaa

bababaa

baababa

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

2

15

4

152

2

15

4

1524

2

15

4

1524

2

15

4

12324

2

153

4

34

2

153

4

34

2

143122

4

34

2

143122

4

34

222

22

22

222

222

2222

2222

xxxxx

xxx

xxx

xxxxx

xxxxx

xxxxxx

xxxxxx

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

6 9 3 5 221

5

7

15

2 2 2 2 2 2 4 2x y xy x y x y xy x y z xy z

22222

2222

222222

249114

91159

992536

xyyxxyxyyx

xyxyyxyx

xyxyxyyxyxyx

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=






1

55

2

3

7

15

5

63

2

3

2 3 2 3 2 3 2 3 2 2 2x y x y x y x y xy xy xy

xyxy

yxxyyx

xyyx

3

10

2

3.5

6

9.

51

57

6

4185

15

710753

2

322

2

332

1

5

232

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

13

762753

627753

62775636

32175236

x

xxxx

xxxx

xxxx

xxxx

###=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

523

523334343

345233433

3432223

22

1

29

10

9

10

10

9

10

9

2

1

10

92

9

10

10

9

2

1

9

10

10

9

5

6

3

2

3

5

4

3

9

10

baba

baababbababa

babaabbabaab

babaabbaab

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

4 23

23

2

3

1

3

1

262 2 2 2 2 2 2x y x xy y x xy xy x y

4 24

3

3

23 2

1

62

24

3

3

22

1

62

24

3

3

2

1

6

28 9 1

62

2 2 4 3 3 2 2 2 2 3 3 3

4 3 3 3 3 3

4 3 3 3

4 3 4

x y x x y x y x y x y xy x y xy

x x y x y xy x y xy

x x y x y x y

x x y x

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=






yxyxyxyyyx

3

2

2

1

3

2

3

42

3

1

3

1

3

1

2

1

2

2

22

22

9

5

2

3

9

41

3

41

3

32

2

21

9

4

9

1

3

4

3

1

3

1

3

2

2

1

9

4

3

1

3

4

3

2

9

1

3

1

2

1

yyxyx

yyxyx

yyyyxyxyxx

yxyxyxyyyx

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

2222222

5

2

6

1

3

2

2

32

4

3

2

1

5

2baaababbaab

abbabababababa

bababab

bababab

bababab

babababaaab

baaababbaab

22222222

22222

22222

22222

2222222

2222222

5

2

6

1

5

2

6

1

5

2

6

1

2

5

12

5

5

2

5

2

6

12

2

5

12

5

5

2

5

2

6

12

2

5

12

5

5

2

5

2

6

12

2

32

12

896

5

2

5

2

6

12

2

3

3

2

4

3

2

1

5

2

5

2

6

1

3

2

2

32

4

3

2

1

5

2

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

babaabccba

2

9

3

4

3

4

5

2

15

2

5

4 2223

babababa

babaabc

cba

2222

2223

15

86

15

86

2

9

3

4

3

4

5

21

2

15

5

4

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

a + 2b ∙ a − 2b ∙ 3b − 2 − 4b2 ∙ (2 − 3b) =

= a2 − 4b2 ∙ 3b − 2 − 8b2 + 12b3 =

= 3a2b − 2a2 − 12b3 + 8b2 − 8b2 + 12b3 =

= 3a2b − 2a2

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=






423633423

2

22 2:22:3

4:

3

2xbaxbaxyyxx

xxxxxxxxxx

xxx

xbaxbaxxx

xbaxbaxxx

xbaxbaxy

yxx

32

13

32

49

8

1

32

9

8

18:

4

9

8

18:

2

3

64

88:

2

12

64:88:2

1

64:88:2

1

64:88:4

3

3

2

343

2

2

3

2

2

812691263

2

22

812691263

2

22

812691263

2

22

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

211132

2416231733

4161732323

4161732323

461015232323

2325231

23

22233

23223

23223

223223

22

xxx

xxxxxxxx

xxxxxxxx

xxxxxxxx

xxxxxxxxxx

xxxxxx

21122 23 xxx

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

bababababaa 511254332 2

2

22

22222

22222

3399

58311209

583112022956

511820253926

bbaba

bbbababa

bbbababababaaaa

babababababaaba

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

xaaxax

xxaaxxax

223

3423534

4615

3

1:

3

425

3

1

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=






yxxyyx 2

3

5

4

3

3

2

4

525

5

4

yx

yx

yyyxxx

yxxyyx

2

3

9

172

2

412

9

1810180

22

12

9

1020

229

10

2

120

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

yxyxyx 64

2

13

3

2

3223

322223

22

22

32163

32368

32163

32128

642

1

3

82

642

13

3

12

yxyyxx

yxyxyyxyxx

yxyxyx

yxyxyxyx

3223 318

3

48 yxyyxx

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

2

2

2222

3

1

2

1

5

3

5

2

2

1

5

1

5

1

4

1

5

2yxxyxyyxyx

3333

33

3333

233323

22323223

25

4

10

1

25

4

20

2

25

22

20

35

20

3

25

2

25

2

4

1

5

1

20

3

25

2

25

2

5

1

4

1

3

1

4

1

5

3

25

2

10

1

25

2

10

1

5

1

4

1

yxyx

yx

xyyx

yxxyyxyx

yxxyyxyyxxyx

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=






baaabbababaabaa 23222

3

1

4

32

4

1

3

1

32234

24

35

48

13bababa

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

xyyxyxyxxyyx 63

3

132

2

1

3

2 2

3223

3

2

3

17

3

4

2

1yxyyxx

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

−2

3𝑥𝑦

1

2𝑥3𝑦 − 2𝑥3𝑦 + 𝑥3𝑦 −𝑥𝑦 −

1

2𝑥𝑦 − 2𝑥4𝑦2 =

= −2

3𝑥𝑦 −

3

2𝑥3𝑦 + 𝑥3𝑦 −

3

2𝑥𝑦 − 2𝑥4𝑦2 =

= 𝑥4𝑦2 −3

2𝑥4𝑦2 − 2𝑥4𝑦2 =

= −3

2𝑥4𝑦2 − 𝑥4𝑦2 = −

5

2𝑥4𝑦2

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

−𝑦2 − 1

2𝑥2 −

3

4𝑥2 − 4𝑦2 + 𝑥𝑦 −

3

2𝑥𝑦 −

2

3𝑥2 ∙

3

5=

= −𝑦2 − 1

2𝑥2 −

3

4𝑥2 + 4𝑦2 − 𝑥𝑦 −

3

2𝑥𝑦 +

2

3𝑥2 ∙

3

5=

= −𝑦2 − 6 − 9 + 8

12𝑥2 +

−2 − 3

2𝑥𝑦 + 4𝑦2 ∙

3

5=

= −𝑦2 − 5

12𝑥2 −

5

2𝑥𝑦 + 4𝑦2 ∙

3

5=

= −𝑦2 −5

12𝑥2 +

5

2𝑥𝑦 − 4𝑦2 ∙

3

5=

= −5

12𝑥2 +

5

2𝑥𝑦 − 5𝑦2 ∙

3

5=

=1

4𝑥2 +

3

2𝑥𝑦 − 3𝑦2

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=






15

62

3

12

5

2 xxx

3

1

3

1

3

1

51

5

3

1

15

2103

5

2

3

1

5

2

15

2

3

2

5

51

6

15

2

3

1

3

2

5

2

5

3

1

5

2

xxxxx

xxx

xxx

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

22322

3222

131222

3

13

2

1222

xxxx

xxx

xxx

xxx

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

xxx

10

7

5

2

2

5

2

5

20

5

2

10

725

5

2

10

7

5

1

2

1

10

7

5

2

5

1

2

1

x

x

xxx

xxx

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Sitografia utilizzata per alcuni esercizi

stringher.blog.kataweb.it/

E-learning IPSSCART B. Stringher – Udine

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=



http://stringher.blog.kataweb.it/




KEYWORDS

Algebra, calcolo letterale, monomio, polinomio, binomio, trinomio, prodotti

notevoli, esercizi con soluzioni

Algebra, Monomial, Polynomial, binomial, trinomial, perfect square trinomials, algebraic factoring, exercises with solution

Algebra, Polinomio, monomio, binomio, trinomio, Igualdades notables, operaciones con polinomios,

Algèbre, Polynôme, Monôme, Polynômes remarquables

Algebra, Polynom, Binom

Dansk (Danish) flerleddet størrelse

Nederlands (Dutch) polynoom betreffend, polynoom,

Français (French) polynôme

Deutsch (German) Polynom

Ελληνική (Greek) πολυώνυμο

Italiano (Italian) polinomio, polinomiale

Português (Portuguese) polinômio (m)

Русский (Russian) многочлен, многочленный

Español (Spanish) polinomio

Svenska (Swedish) polynom (matemat. uttryck i flera led)

中文（简体） (Chinese (Simplified))

多名的, 多项式的, 多词学名, 多项式

中文（繁體） (Chinese (Traditional))

n. - 多詞學名, 多項式

한국어 (Korean)

n. - 다항식

日本語 (Japanese)

n. - 多項式, 多名式学名

يه عرب (Arabic) ال

تعددي الحدود (صفه)متعدد الحدود (االسم)



Operazioni con i polinomi - Istituto Comprensivo Statale ... · Operazioni con i polinomi - 3...

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