The problem states that $x - 2$ is a factor of the polynomial $p(x) = a(x^3 - 2x) + b(x^2 - 5)$. To determine the correct answer, we can use the Factor Theorem, which tells us that if $x - 2$ is a factor of $p(x)$, then $p(2) = 0$.

Step 1: Write out the polynomial expression

Expand the given polynomial: $p(x) = a(x^3 - 2x) + b(x^2 - 5)$ Simplifying this expression: $p(x) = a(x^3 - 2x) + b(x^2 - 5) = ax^3 - 2ax + bx^2 - 5b$ So the full polynomial is: $p(x) = ax^3 + bx^2 - 2ax - 5b$

Step 2: Apply the Factor Theorem

Since $x - 2$ is a factor, $p(2) = 0$. Substituting $x = 2$ into the polynomial: $p(2) = a(2^3) + b(2^2) - 2a(2) - 5b$ $p(2) = a(8) + b(4) - 2a(2) - 5b$ $p(2) = 8a + 4b - 4a - 5b$ $p(2) = (8a - 4a) + (4b - 5b)$ $p(2) = 4a - b$ For $x - 2$ to be a factor, $p(2) = 0$. So we have the equation: $4a - b = 0$ This simplifies to: $b = 4a$

Step 3: Check the options

• A) $a + b = 0$ – Incorrect.
• B) $2a - b = 0$ – Incorrect.
• C) $2a + b = 0$ – Incorrect.
• D) $4a - b = 0$ – Correct.

Thus, the correct answer is D) 4a - b = 0.

Would you like any further details or clarifications?

Here are 5 follow-up questions:

1. What is the Factor Theorem and how is it applied in general cases?
2. Can we apply synthetic division to verify the factor in this case?
3. How would the answer change if the factor was $x + 2$?
4. What are other methods to find factors of polynomials?
5. How would the answer change if $p(x)$ had more terms?

Tip: Always apply the Factor Theorem when testing if a linear expression is a factor of a polynomial.