Question

please answer this question​

Answer 1

Answer:

a = -1 , b = -4

Step-by-step explanation:

f(x) = ax³ + 3x² - bx - 12

f(2) = 8 a + 12 - 2 b - 12 = 0

4 a - b = 0

b = 4 a

f( - 3 ) = - 27 a + 27 + 3b - 12 = 0

- 27 a + 3 b = - 15

- 27a + 3×4a = -15

15 a = - 15

a = - 1

b = - 4

Answer 2

Question:

If (x - 2) and (x + 3) are factors of p(x) = ax³ + 3x² - bx - 12, find the values of a and b

Answer:

a = 1 & b = 4

Step-by-step explanation:

Given polynomial, p(x) = ax³ + 3x² - bx - 12

• (x - 2) is a factor

x - 2 = 0

x = +2

2 is a zero of the polynomial.

Put x = 2, then p(2) = 0

a(2)³ + 3(2)² - b(2) - 12 = 0

a(8) + 3(4) - 2b - 12 = 0

8a + 12 - 2b - 12 = 0

  8a - 2b = 0

  2b = 8a

   b = 8a/2

   b = 4a → [1]

• (x + 3) is a factor

x + 3 = 0

x = -3

-3 is a zero of the polynomial.

Put x = -3, then p(-3) = 0

a(-3)³ + 3(-3)² - b(-3) - 12 = 0

a(-27) + 3(9) + 3b - 12 = 0

-27a + 27 + 3b - 12 = 0

  -27a + 3b + 15 = 0

Put b = 4a, [∵ equation 1]

-27a + 3(4a) + 15 = 0

-27a + 12a + 15 = 0

-15a + 15 = 0

 15a = 15

  a = 15/15

 a = 1

⇒ b = 4a

⇒ b = 4(1)

⇒ b = 4

Therefore, a = 1 and b = 4