Question

If f(x) = x3 + ax2 – 2x + a + 4, then find value of a if x + a is a factor of f(x).​

Answer 1

Step-by-step explanation:

Given :-

f(x) = x^3 + ax^2 – 2x + a + 4

To find :-

Find value of a if x + a is a factor of f(x)?

Solution:-

Given cubic polynomial f(x) = x^3 +ax^2-2x + a + 4

Given that

(x+a) is a factor of f(x).

We know that

Factor Theorem:-

Let P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial, if x-a is a factor of P (x) then P(a) = 0 vice-versa.

Now

We have f(-a) = 0

=> (-a)^3+a(-a)^2-2(-a)+a+4 = 0

=> -a^3 +a(a^2) -(-2a) +a +4 = 0

=> -a^3 + a^3 +2a +a + 4 = 0

=> (-a^3+a^3)+(2a+a) +4 = 0

=> 0+3a +4 = 0

=> 3a+4 = 0

=> 3a = -4

=> a = -4 / 3

Therefore, a = -4/3

Answer:-

The value of a for the given problem is -4/3

Used formulae:-

Factor Theorem:-

Let P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial, if x-a is a factor of P (x) then P(a) = 0 vice-versa.