If p(x) = 2x 3 +ax 2 -7x+b and we want p(1)=3 and p(2) = 19. What should be a and b?

a and b should be a = 3 and b = 5

Step-by-step explanation:

We will make aa system of equations to find a and b

∵ p(x) = 2x³ + ax² - 7x + b

∵ p(1) = 3

→ That means substitute x by 1 and p(x) by 3

∵ 3 = 2(1)³ + a(1)² - 7(1) + b

∴ 3 = 2 + a - 7 + b

→ Add the like terms in the right side

∵ 3 = a + b + (2 - 7)

∴ 3 = a + b + (-5)

∴ 3 = a + b - 5

→ Add 5 to both sides

∴ 3 + 5 = a + b - 5 + 5

∴ 8 = a + b

→ Switch the two sides

∴ a + b = 8 ⇒ (1)

∵ p(2) = 19

→ That means substitute x by 2 and p(x) by 19

∵ 19 = 2(2)³ + a(2)² - 7(2) + b

∴ 19 = 2(8) + a(4) - 14 + b

∴ 19 = 16 + 4a - 14 + b

→ Add the like terms in the right side

∵ 19 = 4a + b + (16 - 14)

∴ 19 = 4a + b + (2)

∴ 19 = 4a + b + 2

→ Subtract 2 both sides

∴ 19 - 2 = 4a + b + 2 - 2

∴ 17 = 4a + b

→ Switch the two sides

∴ 4a + b = 17 ⇒ (2)

Now we have a system of equations to solve it

→ Subtract equation (1) from equation (2)

∵ (4a - a) + (b - b) = (17 - 8)

∴ 3a + 0 = 9

∴ 3a = 9

→ Divide both sides by 3

∴ a = 3

→ Substitute the value of a in equation (1) to find b

∵ 3 + b = 8

→ Subtract 3 from both sides

∴ 3 - 3 + b = 8 - 3

∴ b = 5

∴ a and b should be a = 3 and b = 5

step-by-step explanation: