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)?

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 smallest possible degree of the polynomial f(x) + b\cdot g(x)?

Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1. What is the degree of f(x)*g(x)?

The degree of the polynomial p(x) is 11, and the degree of the polynomial q(x) is 7. Find all possible degrees of the polynomial p(x)+q(x)

Guest Apr 11, 2020

#1**+1 **

A degree of a term is the number of variables in that term (or, if it has only one variable, the exponent of that variable).

Example: x^{4} is of degree 4 because it contains 4 variables (the variable x four times)

-3x^{2} is of degree 2 because it contains 2 variables (the variable x twice); the constant in front has no bearing

upon the degree of the term (unless the constant is zero)

2 (actually, every constant) is of degree zero because it contains no variables

The degree of a polynomial is the largest degree of each of its terms.

Example: f(x) = x^{4} - 3x^{2} + 2

has three terms: x^{4} (which is of degree 4; -3x^{2} (which is of degree 2); and 2 (which is of degree 0)

choose the term with the highest degree (x^{4}) -- so the degree of this function is 4

the other terms have no effect upon this choice.

Problem #1: f(x) = x^{4} - 3x^{2} + 2 is of degree 4

g(x) = 2x^{4} - 6x^{2} + 2x -1 is of degree 2

Multiplying g(x) by a constant won't introduce any new variables, so the degree of f(x) + a·g(x)

won't be larger than the degree of f(x) + g(x) and f(x) + g(x) = (x^{4} - 3x^{2} + 2) + (2x^{4} - 6x^{2} + 2x -1)

= 3x^{4} - 3x^{2} - 2x + 1 (which is of degree 4).

However, it could be smaller -- if a = -½, then the x^{4} term would cancel ...

Problem #2: I don't understand "b\cdot g(x)"

Problem #3: the degree of f(x) * g(x) can be found by multiplying the two functions together.

Since we want to find the degree of the largest degree term, all we need to do is to multiply the term

of f(x) with the highest degree times the term of g(x) with the highest degree:

x^{4} · 2x^{4} = 2x^{8} -- so we now have a function of degree 8.

If you multiply all of the terms of both functions, there won't be any term with a degree higher than 8.

Problem #4: If p(x) has degree 11 and q(x) has degree 7, there will be only one possible degree of

p(x) + q(x); it will always be 11.

geno3141 Apr 11, 2020