Introduction to Factor Theorem and Polynomial Division

In this article, we explore the process of finding the value of (a) such that (x-2) is a factor of the polynomial (x^3-3x 5a). We will use the Factor Theorem and polynomial division to determine the value of (a). Let's dive into the steps to solve this problem.

Step-by-Step Solution

We start with the given conditions and the polynomial expression (x^3 - 3x 5a). We need to find the value of (a) such that (x-2) is a factor of this polynomial. Given:

If (x 2), then (x^3 - 3x 5a 0).

We substitute (x 2) into the equation:

(2^3 - 3(2) 5a 0)

(8 - 6 5a 0)

(2 5a 0)

(5a -2)

(a -frac{2}{5})

Alternatively, we can use the Factor Theorem and polynomial division to solve for (a). According to the Factor Theorem, if (x-2) is a factor of (x^3 - 3x 5a), then substituting (x 2) into the polynomial should yield 0. Let's divide (x^3 - 3x 5a) by (x-2) and express the quotient as a quadratic polynomial:

Let (f(x) x^3 - 3x 5a)

Let (f(x) (x-2)q(x))

Let (q(x) px^2 qx t)

Then,

(x^3 - 3x 5a (x-2)(px^2 qx t))

Expanding the right-hand side:

(x^3 - 3x 5a px^3 - 2px^2 qx^2 - 2qx tx - 2t)

Collecting like terms:

(x^3 - 3x 5a px^3 (q - 2p)x^2 (-2q t)x - 2t)

By comparing the coefficients on both sides of the equation, we get:

1. (p 1)

2.

3.

4.

Solving these equations:

From 2: implies which gives

From 3: implies which gives

From 4: implies which gives

Conclusion

The value of (a) such that (x-2) is a factor of (x^3 - 3x 5a) is (-frac{2}{5}). This solution is derived using both direct substitution and polynomial division methods, ensuring the accuracy and robustness of the result.

Additional Notes

- **Factor Theorem**: The theorem states that if (x - c) is a factor of a polynomial (P(x)), then (P(c) 0). This is a fundamental concept in algebra and is widely used in solving problems involving polynomials. - **Polynomial Division**: Dividing polynomials is a key skill in algebra, and it helps in understanding the structure and factors of polynomials. - **Algebraic Factorization**: The process of expressing a polynomial as a product of simpler polynomials is crucial in solving various algebraic problems. It involves using techniques such as factoring by grouping, special products, and synthetic division.