One of the factors of 3x^3+36x^2+105x is x+b where b is a positive constant. What is the smallest possible value of b? square

Question

Answer 1

To find the smallest possible value of

for which

is a factor of

, we need to use the Factor Theorem. According to the Factor Theorem, if

is a factor of the polynomial \( P(x) \), then \( P(-b) = 0 \).Given \( P(x) = 3x^3 + 36x^2 + 105x \), we need to find

such that \( P(-b) = 0 \).First, let's factor out the common term

from the polynomial:

Now, we need to find the roots of the quadratic equation

. We can use the quadratic formula

, where

,

, and

.Calculate the discriminant:

Now, find the roots:

This gives us two solutions:

Since

must be positive, we take the negative of these roots:

The smallest possible value of

is:

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