Question

question 3 of 5
select the correct answer.
let f(x) and g(x) be polynomials as shown below.
f(x)=a0 + a1x + a2x^2...+anx^n
g(x)=b0 + b1x + b2x^2...+bmx^m
which of the following is true about f(x) and g(x)?
f(x) and g(x) are not closed under multiplication because when multiplied, the result will be a
f(x) and g(x) are not closed under multiplication because when multiplied, the result will not be
f(x) and g(x) are closed under multiplication because when multiplied, the result will not be a po
f(x) and g(x) are closed under multiplication because when multiplied, the result will be a polyno

Explanation:

Step1: Recall polynomial multiplication property

When we multiply two polynomials \(f(x)=\sum_{i = 0}^{n}a_{i}x^{i}\) and \(g(x)=\sum_{j=0}^{m}b_{j}x^{j}\), we use the distributive property. The product \(f(x)g(x)=\sum_{i = 0}^{n}\sum_{j = 0}^{m}a_{i}b_{j}x^{i + j}\).

Step2: Analyze the result

The result of \(f(x)g(x)\) is also a polynomial. A set is closed under an operation if performing that operation on elements of the set results in an element of the set. Since the product of two polynomials \(f(x)\) and \(g(x)\) is a polynomial, polynomials are closed under multiplication.

Answer:

The fourth option: \(f(x)\) and \(g(x)\) are closed under multiplication because when multiplied, the result will be a polynomial.