Part a: Let f(x)=x^4-3x^2+2 and
g(x)=2x^4-6x^2+2x-1. Let a be a constant. What is the largest possible degree of f(x)+a*g(x)?
Part b: Let x^4-3x^2+2 and
g(x)=2x^4-6x^2+2x-1. Let b be a constant. What is the smallest possible degree of the polynomial f(x) + b*g(x)?
(a)
f and g are both degree 4, so their sum cannot be of higher degree.
(b)
1/2 g(x) = x^4-3x^2+x - 1/2
f - 1/2 g = -x + 5/2
Looks like 1 is the smallest possible degree.
Part a: To determine the largest possible degree of f(x) + a*g(x), we need to determine the degree of each individual polynomial and then determine the degree after adding them together.
The degree of a polynomial is determined by the highest power of x present in the polynomial. In this case, we have the following polynomials:
f(x) = x^4 - 3x^2 + 2 (degree 4)
g(x) = 2x^4 - 6x^2 + 2x - 1 (degree 4)
To find the degree of f(x) + a*g(x), we need to add the powers of x with the highest degree. Since both f(x) and g(x) have a degree of 4, the largest possible degree of f(x) + a*g(x) occurs when the coefficients of x^4 are non-zero in both f(x) and g(x).
Therefore, the largest possible degree of f(x) + a*g(x) is 4.
Part b: To determine the smallest possible degree of the polynomial f(x) + b*g(x), we need to determine the degree of each individual polynomial and then determine the degree after adding them together.
Using the same polynomials from Part a:
f(x) = x^4 - 3x^2 + 2 (degree 4)
g(x) = 2x^4 - 6x^2 + 2x - 1 (degree 4)
To find the degree of f(x) + b*g(x), we need to add the powers of x with the highest degree. Since both f(x) and g(x) have a degree of 4, the smallest possible degree of f(x) + b*g(x) occurs when the coefficients of x^4 are zero in either f(x) or g(x), or both.
In our case, since the coefficient of x^4 in f(x) is non-zero and the coefficient of x^4 in g(x) is also non-zero, the smallest possible degree of f(x) + b*g(x) is 4.
Therefore, the smallest possible degree of f(x) + b*g(x) is 4.