Question

if (x-1) an (x+2) are the factors of f(x)= 3x^3 +ax2-bx +2c , then find the value of a+2b-c

Answer 1

the value of a+2b-c  is 15 if (x-1) an (x+2) are the factors of f(x)=3x³ + ax² - bx + 2c

Given:

• (x-1) an (x+2) are the factors of f(x)
• f(x) = 3x³ + ax² - bx + 2c

To Find:

• Value of a + 2b - c

Solution:

• Factor Theorem. x – a is a factor of the polynomial p(x), if p(a) = 0.
• Also, if x – a is a factor of p(x), then p(a) = 0,
• where a is any real number.

Step 1:

x - 1 is a factor of f(x) hence f(1) = 0

3(1)³ + a(1)² - b(1) + 2c = 0

=> 3 + a - b + 2c = 0

=> a - b + 2c = - 3       Eq1

Step 2:

x + 2 is a factor of f(x) hence f(-2) = 0

3(-2)³ + a(-2)² - b(-2) + 2c = 0

=> -24 + 4a +2b + 2c = 0

=> 4a +2b + 2c =  24

=>  2a + b + c  = 12     Eq2

Step 3:

Eq2 - Eq1

=> a + 2b - c  = 15

Hence,  the value of a+2b-c  is 15