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