+2653 Determine all of the following for $f(x) \cdot g(x)$, where $f(x) = -x^2+ 8x - 5$ and $g(x) = x^3 - 11x^2 + 2x$.
Leading term
Leading coefficient
Degree
Constant term
Coefficient of x^2
+760 Analyzing f(x)⋅g(x)
We are asked to find several properties of the product of two polynomials, $f(x) = -x^2+ 8x - 5$ and $g(x) = x^3 - 11x^2 + 2x$.
Finding the Product:
The product f(x)⋅g(x) can be found using the distributive property or polynomial multiplication techniques. Here, we'll use the distributive property:
f(x) * g(x) = (-x^2 + 8x - 5) * (x^3 - 11x^2 + 2x)
Multiplying each term of f(x) by each term of g(x) and combining like terms will result in a fourth-degree polynomial.
Properties of the Product:
Leading Term:
The leading term in a polynomial is the term with the highest degree. In the product, the highest degree term will come from multiplying the highest degree terms of f(x) and g(x). These terms are -x^2 and x^3, respectively. Their product is -x^5. Therefore, the leading term of f(x) * g(x) is -x^5.
Leading Coefficient:
The leading coefficient is the coefficient of the leading term. In this case, the leading term is -x^5, and its coefficient is -1. Therefore, the leading coefficient is -1.
Degree:
The degree of a polynomial is the highest exponent of the variable. Since the leading term has a degree of 5, the degree of f(x) * g(x) is 5.
Constant Term:
The constant term is the term that doesn't include any variable (x). To find it, we identify the terms in the product that don't have any x. Multiplying the constant term of f(x) (-5) by the constant term of g(x) (2) will result in the constant term of the product. Additionally, there might be other terms that don't have x due to the product of other terms. After multiplying and combining like terms, you'll find the constant term.
Coefficient of x^2:
The coefficient of x^2 is the number multiplying the term with x^2. This can be found by identifying terms in the product that have x^2. Multiply terms in f(x) that contain x by terms in g(x) that contain x^2, and vice versa. Combine like terms to find the coefficient of x^2.
Finding these properties requires multiplying f(x) and g(x) and performing some term manipulation. However, the steps outlined above should guide you in finding each property.
