Answer:
The leading term of a polynomial refers to the term with the highest degree. In order to determine the leading term for the polynomial f(x) = -x^3(x^2-4)(3x+2)^2 without fully multiplying it out, we can use the concept of degree of a polynomial and apply the distributive property. Let's break down the polynomial into its factors: f(x) = -x^3(x^2-4)(3x+2)^2 First, we can identify the degree of each factor: The degree of -x^3 is 3 because the variable x is raised to the power of 3. The degree of (x^2-4) is 2 because the variable x is raised to the power of 2. The degree of (3x+2)^2 is 2 because the variable x is raised to the power of 1 (since 3x can be written as x^1) and then further raised to the power of 2. Next, we can determine the degree of the entire polynomial by finding the sum of the degrees of its factors: Degree of f(x) = degree of -x^3 + degree of (x^2-4) + degree of (3x+2)^2 Degree of f(x) = 3 + 2 + 2 = 7 Therefore, the leading term of f(x) is the term with the highest degree, which is x^7
Explanation: