In Exercises 15–24, divide using the quotient rule.15x⁹/3x⁴

Verified step by step guidance

1. Identify the expression to be divided: \( \frac{15x^9}{3x^4} \).
2. Apply the quotient rule for exponents, which states \( \frac{a^m}{a^n} = a^{m-n} \), to the variable part \( \frac{x^9}{x^4} \).
3. Subtract the exponents of \( x \): \( x^{9-4} = x^5 \).
4. Divide the coefficients: \( \frac{15}{3} = 5 \).
5. Combine the results: \( 5x^5 \).

Key Concepts

Quotient Rule

The quotient rule is a fundamental principle in calculus used to differentiate functions that are expressed as the ratio of two other functions. It states that if you have a function f(x) = g(x)/h(x), the derivative f'(x) can be calculated using the formula f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))². Understanding this rule is essential for solving problems involving division of functions.

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. In the context of the given problem, it involves simplifying the expression by dividing the coefficients and subtracting the exponents of like terms. Mastery of polynomial division is crucial for simplifying expressions and finding derivatives accurately.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often includes combining like terms, factoring, and canceling common factors. In the context of the given problem, simplifying the expression 15x⁹/3x⁴ requires dividing the coefficients (15/3) and applying the laws of exponents to the variable x. This concept is vital for making complex expressions more manageable and easier to work with.