Question

select the correct answer.
let p(x) be a polynomial as shown below, and let b be an integer.
$p(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$
which of the following is true regarding the sum and difference of p(x) and b?

• the sum and difference of p(x) and b are integers.
• the sum and difference of p(x) and b are polynomials.
• the sum of p(x) and b is a polynomial, and the difference is an integer.
• the sum of p(x) and b is an integer, and the difference is a polynomial.

Explanation:

Step1: Recall polynomial definition

A polynomial is a sum of monomials. A constant \(b\) is a degree-0 polynomial.

Step2: Check sum/difference

Adding/subtracting two polynomials gives a polynomial: \(p(x)+b = (a_0+b)+a_1x+\dots+a_nx^n\) (polynomial), \(p(x)-b=(a_0-b)+a_1x+\dots+a_nx^n\) (polynomial).

Step3: Evaluate other options

Sum/difference are not integers (they have \(x\) terms unless \(p(x)\) is constant, which isn't specified). So other options are false.

Answer:

The sum and difference of \(p(x)\) and \(b\) are polynomials.