One of the factors of 2x^3+42x^2+208x is x+b, where b is a positive constant. What is the smallest possible value of b ?

Question

Answer 1

To find the smallest possible value of

such that

is a factor of

, we can use the Factor Theorem. The Factor Theorem states that if

is a factor of a polynomial \( P(x) \), then \( P(-b) = 0 \).Let \( P(x) = 2x^3 + 42x^2 + 208x \). We need to find the smallest positive value of

such that \( P(-b) = 0 \).First, substitute

into \( P(x) \):

Simplify the expression:

Set this equal to zero and solve for

:

Factor out

:

This gives us two factors:

The first factor

gives

, which is not positive. We need to solve the quadratic equation

.Use the quadratic formula

, where

,

, and

:

This gives us two solutions:

Since we are looking for the smallest positive value of

, the smallest possible value is:

Therefore, the smallest possible value of

is

.

Question