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If x-2 is a factor of x^(2)+6x+16b, then...

If x-2 is a factor of `x^(2)+6x+16b`, then b =

A

1

B

0

C

2

D

-1

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the value of \( b \) given that \( x - 2 \) is a factor of the quadratic expression \( x^2 + 6x + 16b \). ### Step-by-Step Solution: 1. **Understanding the Factor Condition:** Since \( x - 2 \) is a factor of the polynomial \( x^2 + 6x + 16b \), it means that when we substitute \( x = 2 \) into the polynomial, the result should be zero. 2. **Substituting \( x = 2 \):** We substitute \( x = 2 \) into the polynomial: \[ (2)^2 + 6(2) + 16b = 0 \] 3. **Calculating the Expression:** Now, calculate the left side: \[ 4 + 12 + 16b = 0 \] Simplifying this gives: \[ 16 + 16b = 0 \] 4. **Isolating \( b \):** Next, we isolate \( b \): \[ 16b = -16 \] Dividing both sides by 16: \[ b = -1 \] 5. **Conclusion:** Thus, the value of \( b \) is \( -1 \). ### Final Answer: \[ b = -1 \]
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