Question

Find the sum of f(x) and g(x) if f(x) = 5x^4 + 2x^3 + 3x^2 + 1 and g(x) = 7x^3 + 4x^2 + 3 .

a. Is f(x) a polynomial? If not, give an explanation.

b. Is g(x) a polynomial? If not, give an explanation.

c. Is the sum a polynomial? If not, give an explanation.

d. If the sum is a polynomial, identify the leading term.

Answer 1

The sum of the functions f(x) and g(x) is h(x) = 5x^4 + 9x^3 + 7x^2 + 4. Both f(x) and g(x) are polynomials, as well as their sum h(x). The leading term of the sum is 5x^4.

Step-by-step explanation:

To find the sum of two functions, f(x) and g(x), you simply add together the terms with like powers of x. Let's start with the given functions:

f(x) = 5x^4 + 2x^3 + 3x^2 + 1

g(x) = 7x^3 + 4x^2 + 3

The sum of f(x) and g(x) is:

h(x) = f(x) + g(x) = (5x^4 + 2x^3 + 3x^2 + 1) + (7x^3 + 4x^2 + 3)

Combine like terms:

h(x) = 5x^4 + (2x^3 + 7x^3) + (3x^2 + 4x^2) + (1 + 3)

h(x) = 5x^4 + 9x^3 + 7x^2 + 4

Regarding the questions:

• a. Yes, f(x) is a polynomial because it is a sum of non-negative integer powers of x.
• b. Yes, g(x) is a polynomial for the same reason as f(x).
• c. The sum, h(x), is also a polynomial as it retains the structure of a polynomial.
• d. The leading term of the sum, h(x), is the term with the highest power, which is 5x^4.