Sumativa 4 - Problemas propuestos PDF

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2. Polinomio por polinomio 21. (5𝑥4 𝑦 − 3𝑥2 𝑦 3 − 6𝑥𝑦 )(3𝑥4 𝑦 − 4𝑥2 𝑦 3 + 3𝑥𝑦 ) 5𝑥4 𝑦 − 3𝑥2 𝑦 3 − 6𝑥𝑦 3𝑥4 𝑦 − 4𝑥2 𝑦 3 + 3𝑥𝑦 15𝑥8 𝑦 2 − 9𝑥6 𝑦 4 − 18𝑥5 𝑦 2 −20𝑥6 𝑦 4 + 12𝑥4 𝑦 6 + 24𝑥³𝑦⁴ 5 2 +15𝑥 𝑦 − 9𝑥3 𝑦 4 + 18𝑥2 𝑦 2 𝑹 = 𝟏𝟓𝒙𝟖 𝒚𝟐 − 𝟐𝟗𝒙𝟔 𝒚𝟒 − 𝟑𝒙𝟓 𝒚𝟐 + 𝟏𝟐𝒙𝟒 𝒚𝟔 + 𝟏𝟓𝒙𝟑 𝒚𝟒 − 𝟏𝟖𝒙𝟐 𝒚𝟐 2

1

5

5

1

5

2

3

1

5

22. ( 2 𝑚 2 − 3𝑚𝑛 + 3 𝑛2 ) ( 3 𝑚 − 2 𝑛)= 3 𝑚 3 − 2𝑚2 𝑛 + 9 𝑚𝑛2 − 4 𝑚 2 𝑛 + 2 𝑚𝑛2 − 6 𝑛3 = 𝑚 3 − 3 𝑹=

𝟏𝟑 𝟒

𝒎𝟐 𝒏 +

𝟏

𝟑𝟏

𝒎𝒏𝟐 − 𝒏𝟑 𝟔

𝟏𝟖

23. (2𝑥𝑎+3 + 5𝑥𝑎+2 − 𝑥 𝑎+1 + 𝑥𝑎−2 )(𝑥𝑎+1 + 2𝑥𝑎 − 𝑥𝑎−1 ) 2𝑥𝑎+3 + 5𝑥𝑎+2 − 𝑥𝑎+1 + 𝑥 𝑎−2 +𝑥𝑎+1 + 2𝑥𝑎 − 𝑥𝑎−1 2𝑥2𝑎+4 + 5𝑥 2𝑎+3 − 𝑥2𝑎+2 + 𝑥2𝑎 2𝑎+3 2𝑎+2 2𝑎 +1 +4𝑥 + 10𝑥 − 2𝑥 + 2𝑥2𝑎−2 2𝑎+2 2𝑎+1 −2𝑥 − 5𝑥 − 𝑥2𝑎 − 3 𝟐𝒂+𝟒 𝟐𝒂+ 𝟐𝒂+𝟐 𝟐𝒂+𝟏 𝟐𝒂 𝟐𝒂−𝟏 𝟐𝒂−𝟐 𝑹 = 𝟐𝒙 + 𝟗𝒙 + 𝟕𝒙 −𝟕𝒙 +𝒙 +𝒙 + 𝟐𝒙 − 𝒙𝟐𝒂−𝟑 24. (4𝑥3 − 2𝑥 2 𝑦 + 6𝑥𝑦 2 )(𝑥 2 𝑦 − 𝑥𝑦 2 − 2𝑦3 ) = 4𝑥5 𝑦 − 4𝑥 4 𝑦 2 − 8𝑥3 𝑦 3 − 2𝑥4 𝑦 2 + 2𝑥3 𝑦 3 + 4𝑥2 𝑦 4 + 6𝑥 3 𝑦 3 − 6𝑥 2 𝑦 4 − 12𝑥𝑦 5 𝑹 = 𝟒𝒙𝟓 𝒚 − 𝟔𝒙𝟒 𝒚𝟐 − 𝟐𝒙𝟐 𝒚𝟒 − 𝟏𝟐𝒙𝒚𝟓 25.(2𝑥2𝑚+1 + 3𝑥 2𝑚 − 𝑥2𝑚−1 )(𝑥2 + 2𝑥 + 1) 2𝑥2𝑚+3 + 4𝑥2𝑚+2 + 2𝑥2𝑚+1 + 3𝑥2𝑚+2 + 6𝑥 2𝑚+1 + 3𝑥2𝑚 − 𝑥2𝑚+1 − 2𝑥2𝑚 − 𝑥2𝑚−1 𝑹 = 𝟐𝒙𝟐𝒎+𝟑 + 𝟕𝒙𝟐𝒎+𝟐 + 𝟕𝒙𝟐𝒎+𝟏 + 𝒙𝟐𝒎 − 𝒙𝟐𝒎−𝟏 3. Divisiones Algebraicas 3.1. Monomio entre monomio 26.

27.

28.

−10𝑥 7𝑦 6 𝑐 −6𝑥 2𝑦 2𝑐

−10

=

8𝑥 3𝑎−1𝑦 5𝑎−4

2𝑥 2𝑎+3𝑦 3𝑎−1

6

=

8

−20𝑥 5𝑚−2𝑦 9𝑛 𝑧 2𝑚 −6𝑥 3𝑦 5𝑧 2

2

𝑥7−2 𝑦 6−2 𝑐1−1 =

−5 3

𝑥5 𝑦 4

𝑥 (3𝑎−1)−(2𝑎+3) 𝑦 (5𝑎−4)−(3𝑎−1) = 4𝑥𝑎−4 𝑦 2𝑎−3

=

−20 6

𝑥 (5𝑚−2)−(3) 𝑦 9𝑛−5 𝑧 2𝑚−2 =

−𝟏𝟎 𝟑

𝒙𝟓𝒎−𝟓 𝒚𝟗𝒏−𝟓𝒛𝟐𝒎−𝟐

29.

−7 𝑚 𝑛 𝑎 𝑏 8 −3 𝑎𝑏2 4

=

−7 8 −3 4

𝑎 𝑚−1 𝑏 𝑛−2 =

𝟕 𝒎−𝟏 𝒏−𝟐 𝒂 𝒃 𝟔

3.2 Polinomio entre monomio 2𝑥 4 −𝑥 2

5𝑥 3

30.

2𝑥 4−5𝑥 3+𝑥 2 −𝑥 2

31.

16𝑥 6𝑦 5 𝑧−12 𝑥 4𝑦 6𝑧 2 +6𝑥 3𝑦 9 −4𝑥 2𝑦

32.

4𝑥 2𝑚+1+8𝑥 3𝑚−2−12𝑥 𝑚+3 6𝑥 𝑚−2

=

3 7 9 2 8 7 4 4 5 𝑥 𝑦 − 𝑥 𝑦 + 𝑥 𝑦 3 3 4 𝑥𝑦 5 15

33. 5

34.

−

=

−𝑥 2

𝑥2

+

=

=

−𝑥 2

= −2𝑥2 + 5𝑥 − 𝑥 0 = −𝟐𝒙𝟐 + 𝟓𝒙 − 𝟏

16𝑥 6 𝑦 5𝑧

4𝑥 2𝑚+1 6𝑥 𝑚−2

3 7 9 𝑥 𝑦 5 4 𝑥𝑦 5 15

−

−4𝑥 2𝑦

−

𝑎 2𝑥 𝑏3𝑦𝑐 4𝑧 +6𝑎 3𝑥 𝑏4𝑦𝑐 5𝑧 −8𝑎 4𝑥 𝑏5𝑦𝑐 6𝑧 1 2𝑥 3𝑦 4𝑧 𝑎 𝑏 𝑐 2

+

2 8 7 𝑥 𝑦 3 4 5 𝑥𝑦 15

=

12𝑥 4𝑦 6 𝑧 2 −4𝑥 2𝑦

8𝑥 3𝑚−2

−

6𝑥 𝑚−2

+

4 4 5 𝑥 𝑦 3 4 5 𝑥𝑦 15

𝑎 2𝑥 𝑏3𝑦 𝑐 4𝑧

1 2𝑥 3𝑦 4𝑧 𝑎 𝑏 𝑐 2

+

+

6𝑥 3𝑦 9

−4𝑥 2𝑦

12𝑥 𝑚+3

=

6𝑥 𝑚−2

𝟗

𝟒

=

= −𝟒𝒙𝟒 𝒚𝟒 𝒛 + 𝟑𝒙𝟐 𝒚𝟓𝒛𝟐 −

𝟐𝒙𝒎+𝟑

+

𝟑

𝟒𝒙𝟐𝒎 𝟑

𝟑

𝟐

𝒙𝒚𝟖

− 𝟐𝒙𝟓

𝟓

𝒙𝟔 𝒚𝟒 − 𝒙𝟕 𝒚𝟐 + 𝟓𝒙𝟑 𝟐

6𝑎 3𝑥 𝑏4𝑦𝑐 5𝑧

1 2𝑥 3𝑦 4𝑧 𝑎 𝑏 𝑐 2

−

8𝑎 4𝑥 𝑏5𝑦 𝑐 6𝑧

1 2𝑥 3𝑦 4𝑧 𝑎 𝑏 𝑐 2

= 𝟐 + 𝟏𝟐𝒂𝒙 𝒃𝒚 𝒄𝒛 − 𝟏𝟔𝒂𝟐𝒙 𝒃𝟐𝒚 𝒄𝟐𝒛

3.3 Polinomio entre polinomio 3𝑥 2−5𝑥+2 3𝑥−2

= X - 1 3𝑥 − 2 ÷ 3𝑥2 − 5𝑥 + 2 −3𝑥2 + 2𝑥 −3𝑥 + 2 3𝑥 − 2 0 35.

36.

5𝑎 2−21𝑏 2+8𝑎𝑏 𝑎+3𝑏

=

5𝑎 2−7𝑎𝑏+15𝑎𝑏−21𝑏 2 𝑎+3𝑏

=

𝑎(5𝑎−7𝑏)+3𝑏(5𝑎−7𝑏) 𝑎+3𝑏

=

(5𝑎−7𝑏)(𝑎+36) 𝑎+36

37. 𝑎 2 + 𝑎 + 1 ÷ 𝑎 4 − 𝑎 2 − 2𝑎 − 1 = 𝑎 + 1 ÷ 𝑎 4 − 2𝑎 − 1 = 𝑎 +

38.

𝑥 2+5𝑥𝑦+6𝑦 2 𝑥+2𝑦

=

𝑥 2+2𝑥𝑦+3𝑥𝑦+6𝑦 2 𝑥+2𝑦

=

𝑥(𝑥+2𝑦)+3𝑦(𝑥+2𝑦) 𝑥+2𝑦

=

1

𝑎4

(𝑥+2𝑦)(𝑥+3𝑦) 𝑥+2𝑦

= 𝟓𝒂 − 𝟕𝒃

𝟏

− 2𝑎 − 1 = −𝒂 𝒂𝟒 − 𝟏

= 𝒙 + 𝟑𝒚

39.

𝑛4 +2𝑛2 −48 𝑛2+8

(𝑛2)2 +2𝑛2 −48

=

=

𝑛2+8

40. (𝑎 4 − 𝑎) ÷ (𝑎 − 1) =

(𝑢−6)(𝑢+8) 𝑛2+8

𝑎(𝑎−1)(𝑎 2+𝑎+1) 𝑎−1

=

(𝑛2 −6)(𝑛2 +8) 𝑛2 +8

= 𝒏𝟐 − 𝟔

= 𝑎 (𝑎 2 + 𝑎 + 1) = 𝑎. 𝑎 2 + 𝑎. 𝑎 + 𝑎. 1 = 𝒂𝟑 + 𝒂𝟐 + 𝒂

4. Potencias y Raíces −2

42.

1

(𝑥 2+1) 3 (𝑥 2+1) 6 1 (𝑥 2+1) 2

43.(

6𝑥 3𝑦 −2𝑧 4 −2 ) 3𝑥 −1𝑦 4 𝑧 3

1

44.

5

1

(𝑥 −3𝑦 −1𝑧 2 ) 2 .

=

1 2 1 2

−

= (𝑥2 + 1)−1 =

2

32 (𝑥 −1)2(𝑦 4 )2(𝑧 3 )

62 (𝑥 3)2(𝑦 −2)2(𝑧 4 )2

1

5

26 (𝑚 3)6(𝑛6)6

2−1(𝑚 −2)−1 (𝑛6)−125 (𝑚)5(𝑛)5

1

1

. (𝑥2 𝑦 4 )3 (𝑥−2 𝑦 −3 𝑧 −1 ))3 = (

𝑧

(𝑦 4 )3 (𝑦 −3 𝑧 −1 ))3

17 1 𝑥 6 𝑦2

.

9𝑥 −2𝑦 8 𝑧 6

=

36𝑥 6 𝑦 −4𝑧 8

=

1

4

𝑥 −2−6 𝑦 8+4 𝑧 6−8 =

64𝑚 4 𝑛5 2−1𝑚 2𝑛−6 (32𝑚 5𝑛5 )

=

=

𝒚𝟏𝟐

𝟒𝒙𝟖 𝒛𝟐

𝟒𝒏𝟔

𝒎𝟓

1

) = ((𝑥 −3 𝑦 −1 𝑧 2 ) 2 . (𝑥 2 𝑦 4 )3 (𝑥 −2 𝑦 −3 𝑧 −1 ))3 =

(𝑧2 )2

1

𝟏

𝒙𝟐 +𝟏

1

(𝑥 2𝑦 4 ) 3 3

(𝑥 −2𝑦 −3𝑧 −1)−1

1 1 (𝑥 3)2𝑦 2

(

=

3𝑥 −1 𝑦 4𝑧 3

(2𝑚 −2𝑛6 )−1(2𝑚𝑛)5

1

2 1 3 6 1 2

− +

= ( 6𝑥 3𝑦 −2𝑧4 )2 =

(2𝑚 2 . 𝑛6)

45. ( ((

=

=(

1

𝑧

3 1 𝑥 2𝑦 2

𝑧(𝑦 4 )3𝑧 −1 3 17 7 𝑥 6𝑦 2

) =

−4

1

. (𝑥 3 (𝑦 4 )3 ) (𝑦 −3 𝑧 −1 ))3 = 1

(𝑦4 )3 3 (

) =(

17 7 𝑥 6 𝑦2

1

17 𝑥 6

13 𝑦6

)3 =

1

17 13 (𝑥 6 )3(𝑦 6 )3

=

𝟏

𝟏𝟕 𝟏𝟑 𝒙𝟐 𝒚𝟐...