Introduction

Understanding the factors of a polynomial and using the factor theorem can significantly simplify the process of finding unknown coefficients in a polynomial equation. In this article, we'll explore two different methods to determine the value of (m) in the polynomial (3x^3 - 5x^2 - mx^{12}) given that (3x - 2) and (x^2) are its factors.

Method 1: Substitution

The first method involves using the factor theorem, which states that if ((x - a)) is a factor of a polynomial (f(x)), then (f(a) 0). In this problem, we are given that (3x - 2) is a factor. We can substitute (x frac{2}{3}) into the polynomial to find the value of (m).

Substitute (x frac{2}{3}) into (3x^3 - 5x^2 - mx^{12}):

(3left(frac{2}{3}right)^3 - 5left(frac{2}{3}right)^2 - mleft(frac{2}{3}right)^{12} 0)

Calculating the individual terms:

(3 cdot frac{8}{27} - 5 cdot frac{4}{9} - m cdot frac{4096}{531441} 0)

Simplifying further:

(frac{8}{9} - frac{20}{9} - frac{4096m}{531441} 0)

Multiplying by 531441 to clear the denominators:

(531441 cdot left(frac{8}{9} - frac{20}{9}right) - 4096m 0)

(531441 cdot left(-frac{12}{9}right) - 4096m 0)

(-699841 - 4096m 0)

Solving for (m):

(4096m -699841)

(m frac{-699841}{4096} 16)

Method 2: Polynomial Division

Given that (3x - 2) and (x^2) are factors, their product (3x^2 - 4) must be a factor of the polynomial. To find (m), we can use polynomial division to divide the given polynomial by (3x^2 - 4). The quotient should be (x - 3) and the remainder should be 0 (since (3x^2 - 4) divides the polynomial).

Divide (3x^3 - 5x^2 - mx^{12}) by (3x^2 - 4) using synthetic division or polynomial long division:

1. The dividend (3x^3 - 5x^2 - mx^{12}) and the divisor (3x^2 - 4).

2. The quotient is (x - 3) and the remainder is (x^{16} - m).

Since the polynomial is divisible by (3x^2 - 4), the remainder must be 0:

(x^{16} - m 0)

Since (x 0) is not a solution, we have:

(m 16)

Conclusion

Both methods yield the same result: (m 16). The factor theorem and polynomial division are powerful tools in solving polynomial equations. By applying these principles, we can efficiently determine unknown coefficients in polynomials.