Final answer:
Dividing a polynomial by another polynomial, such as 3x-4, entails a process akin to long division. A negative exponent signifies the reciprocal of the base to the positive exponent, aiding the simplification.
Step-by-step explanation:
When a polynomial is divided by 3x-4, we are performing polynomial division, which is a process similar to long division with numbers. This mathematical procedure can be used to simplify expressions, solve algebraic equations, or find the remainder when one polynomial is divided by another. The division may result in a quotient and a remainder. The division process involves finding a term that, when multiplied by the divisor (in this case 3x-4), will match the leading term of the polynomial being divided. This matched term is then subtracted from the polynomial, and the process is repeated until no terms are left to be divided cleanly by 3x-4.
To understand the concept of inversion or dividing by a variable with a negative exponent, we recognize that a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, x-n = 1/xn, which demonstrates the idea of an expression such as 3-4 being equal to 1/34 using the rule of exponents that tells us to add exponents when multiplying same bases. This is evident from expressions like 34·3-4 = 30 = 1, where any number (except zero) to the power of zero is 1.
When handling algebraic expressions and working with polynomials, it is essential to apply these exponent rules, whether the calculation is done by hand or using a calculator to find precise solutions or simplify expressions. Once the polynomial is fully divided by 3x-4, the result is often represented as a linear equation if the polynomial's degree is two. The process not only applies to division but it is also used in solving equations where simplifying the denominator is needed.