Question

Dado o polinômio P(x) = (a-b)x⁴ + ax² + 2x - b - 1, determine a e b de tal forma que P(2) = 18 e P(1) = 5. Calcule P(2).

Answer 1

Resposta passo a passo:

P(x) = (a-b)x⁴ + ax² + 2x - b - 1

a = ?

b = ?

P(2) = 18

P(1) = 5

P(2) = (a-b) • 2⁴ + a • 2² + 2 • 2 - b - 1

18 = (a-b) • 16 + a • 4 + 4 - b - 1

18 = 16(a-b) + 4a + 3 - b

18 - 3 = 16(a-b) + 4a - b

16(a-b) + 4a - b = 15

16a + 16(-b) + 4a - b = 15

16a - 16b + 4a - b = 15

20a - 16b - b = 15

20a - 17b = 15

P(1) = (a-b) • 1⁴ + a • 1² + 2 • 1 - b - 1

5 = (a-b) • 1 + a • 1 + 2 - b - 1

5 = a - b + a + 1 - b

5 - 1 = a - b + a - b

4 = 2a - b - b

4 = 2a - 2b

4 = 2(a - b)

4 / 2 = a - b

a - b = 2

- b = - a + 2

- b = -(a - 2)

b = a - 2

20a - 17b = 15

20a - 17(a - 2) = 15

20a - 17 • a - 17 • (-2) = 15

20a - 17a + 34 = 15

3a + 34 = 15

3a = 15 - 34

3a = - 19

a = - 19 / 3

b = a - 2

b = (- 19 / 3) - 2

b = (- 19 / 3) - 2 • 3 / 3

b = (- 19 / 3) - 6 / 3

b = (- 19 - 6) / 3

b = - (19 + 6) / 3

b = - 25 / 3

P(2) = (a-b) • 2⁴ + a • 2² + 2 • 2 - b - 1

P(2) = (a-b) • 16 + a • 4 + 4 - b - 1

P(2) = 16(a-b) + 4a + 3 - b

P(2) = 16((- 19 / 3) - (- 25 / 3)) + 4(- 19 / 3) + 3 - (- 25 / 3)

P(2) = 16((- 19 / 3) + (25 / 3)) - 76 / 3 + 3 + 25 / 3

P(2) = 16((- 19 + 25) / 3) - 76 / 3 + (3 • 3) / 3 + 25 / 3

P(2) = 16(6 / 3) - 76 / 3 + 9 / 3 + 25 / 3

P(2) = 16 • 2 + (- 76 + 9 + 25) / 3

P(2) = 32 + (- 42) / 3

P(2) = 32 - (42 / 3)

P(2) = 32 - 14

P(2) = 18