§ 14. Сложение и вычитание многочленов

№ 390

Найдите P(a) = P1(a) + P2(a), если

а) P1(a) = 2a + 5; P2(a) = 3a – 7; P(a) = 2a + 5 + 3a – 7 = 5a – 2;

б) P1(a) = 7 – 2a; P2(a) = – 1 – 5a; P(a) = 7 – 2a – 1 – 5a = – 7a + 6;

в) P1(a) = 3a – 4; P2(a) = 11 – 3a; P(a) = 3a – 4 + 11 – 3a = 7;

г) P1(a) = – 4 – 3a; P2(a) = 7 – 8a; P(a) = – 4 – 3a + 7 – 8a = 11a + 3.

№ 391

Найдите P(x) = P1(x) + P2(x), если :

а) P1(x) = 2x

3

+ 5; P2(x) = 3x

3

+ 7; P(x) = 2x

3

+ 5 + 3x

3

+ 7 = 5x

3

+ 12;

б) P1(x) = 6x

2

– 4; P2(x) = 5x

2

– 10; P(x) = 6x

2

– 4 + 5x

2

– 10 = 11x

2

– 14;

в) P1(x)=4x

5

+2x+1; P2(x)=x

5

+x–2; P(x)=4x

5

+2x+1+x

5

+x–2=5x

5

+ 3x – 1;

г) P1(x)=x

11

+x

6

–3; P2(x)=2x

11

+3x

6

+1; P(x)=x

11

+x

6

–3+2x

11

+3x

6

+1=3x

11

+4x

6

– 2.

№ 392

Найдите P(a,b) = P1(a,b) +P2(a,b)

а) P1(a,b) = a + 3b; P2(a,b) = 3a – 3b;P(a,b) = a + 3b + 3a – 3b = 4a;

б) P1(a,b) = a2

– 5ab – 3b2

; P2(a,b) = a2

+ b2

; 58

P(a,b) = a2

– 5ab – 3b2

+ a2

+ b2

= 2a2

– 5ab – 2b2

;

в) P1(a,b) = 8a3

+ 3a2

b – 5ab2

+ b3

; P2(a,b) = 18a3

– 3a2

b – 5ab2

+ 2b3

;

P(a,b) = 8a3

+3a2

b–5ab2

+ b3

+ 18a3

– 3a2

b – 5ab2

+ 2b3

=26a3

– 10a2

b +3b3

;

г) P1(a,b) = 10a4

– 7a3

b – a2

b2

+ 6; P2(a,b) = 17a4

– 10a3

b + a2

b2

+ 3;

P(a,b) =10a4

– 7a3

b – a2

b2

+ 6 + 17a4

– 10a3

b + a2

b2

+ 3 = 27a4

– 17a3

b + 9.

№ 393

Найдите P(y) = P1(y) – P2(y), если:

а) P1(y) = 2y

2

+ 8y – 11; P2(y) = 3y

3

– 6y + 3;

P(y) = 2y

2

+ 8y – 11 – 3y

3

+ 6y – 3 = – 3y

3

+ 2y

2

+ 14y – 14;

б) P1(y) = 4y

4

+ 4y

2

– 13; P2(y) = 4y

4

– 4y

2

+ 13;

P(y) = 4y

4

+ 4y

2

– 13 – 4y

4

+ 4y

2

– 13 = 8y

2

– 26;

в) P1(y) = y

3

– y + 7; P2(y) = y

3

+ 5y +11;

P(y) = y

3

– y + 7 – y

3

– 5y – 11 = – 6y – 4;

г) P1(y) = 15 – 7y

2

; P2(y) = y

3

– y

2

– 15;

P(y) = 15 – 7y

2

– y

3

+ y

2

+ 15 = – y

3

– 6y

2

+ 30.

№ 394

Найдите P(c,d) = P1(c,d) – P2(c,d), если:

а) P1(c,d) = 3c

2

+ d; P2(c,d) = 2c

2

– 3d;

P(c,d) = 3c

2

+ d – 2c

2

+ 3d = c

2

+ 4d;

б) P1(c,d) = 5c

4

+ 3c

2

d – d3

; P2(c,d) = 2c

2

– 3c

2

d + d2

;

P(c,d) = 5c

4

+ 3c

2

d – d3

– 2c

2

+ 3c

2

d – d2

= 5c

4

+ 6c

2

d – d3

– d2

– 2c

2

;

в) P1(c,d) = 12c

2

d – 3cd2

+ 4; P2(c,d) = 6c

2

d – 5cd2

+ 2c;

P(c,d) = 12c

2

d – 3cd2

+ 4 – 6c

2

d + 5cd2

– 2c = 6c

2

d + 2cd2

– 2c + 4;

г) P1(c,d) = c

2

+ 2cd + d2

; P2(c,d) = 5c

2

– 6cd – 7d2

;

P(c,d) = c

2

+ 2cd + d2

– 5c

2

+ 6cd + 7d2

= – 4c

2

+ 8cd + 8d2

.

№ 395

а) (5x – 3) + (7x – 4) = 8 – (15 – 11x);

5x – 3 + 7x – 4 = 8 – 15 + 11x; 12x – 11x = – 7 + 7; x = 0.

Ответ: 0.

б) (4x + 3) – (10x + 11) = 7 + (13 – 4x); 4x + 3 – 10x – 11 = 7 + 13 – 4x;

4x – 10x + 4x = 20 + 11 – 3; – 2x = 28; x = – 14.

Ответ: –14.

в) (7 –10x) – (8 – 8x) + (10x +6) = – 8; 7 – 10x – 8 + 8x + 10x + 6 = – 8;

8x = – 8 – 6 + 8 – 7; x = –

13

8

; x = – 1

5

8

.

Ответ: – 1

5

8

.

г) (2x + 3) + ( 3x + 4) + (5x + 5 ) = 12 – 7x;

2x + 3 + 3x + 4 + 5x + 5 = 12 – 7x;

10x + 7x = 12 – 12; 17x = 0; x = 0.

Ответ: 0. 59

№ 396

а)

3

4

y –

5

1, 25

6

y

⎛⎞

− ⎜⎟

⎝⎠

= 0,55;

9

12

y –

10

12

y + 1,25 = 0,55;

–

1

12

y = 0,55 – 1,25; –

1

12

y = – 0,7; y = 8,4.

Ответ: 8,4.

б)

3

8

x –

1

2,4

3

x

⎛⎞

− ⎜⎟

⎝⎠

= – 0,4;

9

24

x –

8

24

x + 2,4 = – 0,4;

1

24

x = – 0,4 – 2,4;

1

24

x = – 2,8; x = – 67,2.

Ответ: – 67,2.

в)

3

4

x – (0,25x – 3) = 1,2; 0,75x – 0,25x + 3 = 1,2; 0,5x = – 1,8; x = – 3,6.

Ответ: – 3,6.

г)

1

2

x – (2,5x – 3) = 1,8; 0,5x – 2,5x +3 = 1,8; – 2x = – 1,2; x = 0,6.

Ответ: 0,6.

№ 397

а) x – 1; б) (x – 0,5) + (x – 1) + (x – 1,5) = 3x –3;

в) x + x – 0,5 = 2x – 0,5; г) x + (x – 0,5) + (x – 1) + (x – 1,5) = 4x – 3.

№ 398

P1(a) = 2a3

+ 3a3

– a + 1; P2(a) = 4a4

+ 6a3

– 2a2

+ 2a;

P3(a) = 2a5

+ 3a4

– a3

+ a2

;

а) P1(a) + P2(a) + P3(a) = 2a3

+ 3a3

– a + 1 + 4a4

+ 6a3

– 2a2

+ 2a +

+ 2a5

+ 3a4

– a3

+ a2

= 7a3

+ 2a2

+ a + 1 + 7a4

+ 2a5

;

б) P1(a) – P2(a) + P3(a) = 2a3

+ 3a3

– a + 1 – 4a4

– 6a3

+ 2a2

– 2a +

+ 2a5

+ 3a4

– a3

+ a2

= 2a5

– a4

– 5a3

+ 6a2

– 3a + 1;

в) P1(a) + P2(a) – P3(a) = 2a3

+ 3a3

– a + 1 + 4a4

+ 6a3

– 2a2

+ 2a –

– 2a5

– 3a4

+ a3

– a2

= – 2a5

– 3a4

+ a3

– a2

;

г) P1(a) – P2(a) – P3(a) = 2a3

+ 3a3

– a + 1 – 4a4

– 6a3

+ 2a2

– 2a – 2a5

–

– 3a4

+ a3

– a2

= – 2a5

– 7a4

– 3a3

+ 4a2

– 3a + 1.

№ 399

P1(x,y) = 27x

3

– 27x

2

y + 9xy

2

– y

3

; P2(x,y) = 20x

3

– 15x

2

y + 4xy

2

– 3y

3

;

P3(x,y) = 10x

3

+ 12x

2

y – 5xy

2

+ y

2

;

а) P(x,y) = P1(x,y) + P2(x,y) + P3(x,y) = 27x

3

– 27x

2

y + 9xy

2

– y

3

+

+ 20x

3

– 15x

2

y + 4xy

2

– 3y

3

+ 10x

3

+ 12x

2

y – 5xy

2

+ y

2

=

= 67x

3

– 30x

2

y + 8xy

2

– 4y

3

+ y

2

;

б) P(x,y) = P1(x,y) – P2(x,y) + P3(x,y) = 27x

3

– 27x

2

y + 9xy

2

– y

3

–

– 20x

3

+ 15x

2

y – 4xy

2

+ 3y

3

+ 10x

3

+ 12x

2

y – 5xy

2

+ y

2

=

= 17x

3

+ 10x

2

y + 2y

3

+ y

2

; 60

в) P(x,y) = P1(x,y) + P2(x,y) – P3(x,y) = 27x

3

– 27x

2

y + 9xy

2

– y

3

+

+ 20x

3

– 15x

2

y + 4xy

2

– 3y

3

– 10x

3

– 12x

2

y + 5xy

2

– y

2

=

= 37x

3

– 44x

2

y + 18xy

2

– 4y

3

– y

2

;

г) P(x,y) = P1(x,y) – P2(x,y) – P3(x,y) = 27x

3

– 27x

2

y + 9xy

2

– y

3

–

– 20x

3

+ 15x

2

y – 4xy

2

+ 3y

3

– 10x

3

– 12x

2

y + 5xy

2

– y

2

=

= – 3x

3

– 24x

2

y + 10xy

2

+ 2y

3

– y

2

.

№ 400

а) 2x

2

– (2x

2

– 5x) – (4x – 2) = 5;

2x

2

– 2x

2

+ 5x – 4x + 2 = 5; x = 5 – 2; x = 3.

Ответ: 3.

б) 3y

3

– (3y

3

– 6y) – (3y + 4) = 2;

3y

3

– 3y

3

+ 6y – 3y – 4 = 2; 3y = 6; y =2.

Ответ: 2.

в) (x

2

– 7x – 11) – (5x

2

– 13x – 18) = 16 – 4x

2

;

x

2

– 7x – 11 – 5x

2

+ 13x + 18 = 16 – 4x

2

;

x

2

– 5x

2

+ 4x

2

– 7x + 13x = 16 – 18 +11; 6x = 9; x = 1,5.

Ответ: 1,5.

г) (y

2

– 5y – 19) – (5y

2

– 6y – 9) = 26 – 4y – 4y

2

;

y

2

– 5y – 19 – 5y

2

+ 6y + 9 = 26 – 4y – 4y

2

;

y

2

– 5y

2

+ 4y

2

– 5y + 6y + 4y = 26 – 9 + 19; 5y = 36; y =7,2.

Ответ: 7,2.

№ 401

Второй столбец

4x + 1 а) 9x + 7 – 5x – 6 = 4x + 1;

0 б) a3

+ 2a2

b + b3

– a3

– 2a2

b – b3

= 0;

– 4mn в) m2

– 2mn + n2

– m2

– 2mn – n2

= – 4mn;

– 2c

2

d – 3cd2

+ 8 г) 0 – 2c

2

d – 3cd2

+ 8 = – 2c

2

d – 3cd2

+ 8.

№ 402

а) 6a2

– (2 – (1,56a – (a2

+ 0,36a)) + (5,5a2

+ 1,2a – 1)) =

= 6a2

– (2 – 1,56a + a2

+ 0,36a + 5,5a2

+ 1,2a – 1) =

= 6a2

– a2

– 5,5a2

+ 1,56a – 0,36a – 1,2a – 2 + 1 = 0,5a2

– 1;

б) (a2

+ 2x

2

) – (5a2

– 1,2ax + (2,8x

2

– (1,5a2

– 0,5ax + 1,8x

2

))) =

= a2

+ 2x

2

– (5a2

– 1,2ax + (2,8x

2

– 1,5a2

+ 0,5ax – 1,8x

2

)) =

= a2

+ 2x

2

– (5a2

– 1,2ax + 2,8x

2

– 1,5a2

+ 0,5ax – 1,8x

2

) =

= a2

+ 2x

2

– 5a2

+ 1,2ax – 2,8x

2

+ 1,5a2

– 0,5ax + 1,8x

2

=

= a2

–5a2

+1,5a2

+ 2x

2

– 2,8x

2

+ 1,8x

2

+ 1,2ax – 0,5ax=– 2,5a2

+ x

2

+ 0,7ax;

в) 12,5x

2

+ y

2

– (8x

2

– 5y

2

– ( – 10x

2

+ (5,5x

2

– 6y

2

))) =

= 12,5x

2

+ y

2

– (8x

2

– 5y

2

– ( – 10x

2

+ 5,5x

2

– 6y

2

)) =

= 12,5x

2

+ y

2

– (8x

2

– 5y

2

+ 10x

2

– 5,5x

2

+ 6y

2

) =

= 12,5x

2

+ y

2

– 8x

2

+ 5y

2

– 10x

2

+ 5,5x

2

– 6y

2

=

= 12,5x

2

– 8x

2

– 10x

2

+ 5,5x

2

+ y

2

+ 5y

2

– 6y

2

= 0; 61

г) (y

3

+ 3z

2

) – (y

3

– 6az + (2y

3

– (3z

2

+ 4az – 1,2y

3

))) =

= y

3

+ 3z

2

– (y

3

– 6az + (2y

3

– 3z

2

– 4az + 1,2y

3

)) =

= y

3

+ 3z

2

– (y

3

– 6az + 2y

3

– 3z

2

– 4az + 1,2y

3

) =

= y

3

+ 3z

2

– y

3

+ 6az – 2y

3

+ 3z

2

+ 4az – 1,2y

3

=

= y

3

– y

3

– 2y

3

– 1,2y

3

+3z

2

+ 3z

2

+6az + 4az = – 3,2y

3

+ 6z

2

+ 10az.