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+cdots+a_nx^n) which of the following is true regarding the sum and difference of p(x) and b? the sum of p(x) and b is an integer, and the difference is a polynomial. 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 and difference of p(x) and b are integers.

Explanation:

Step1: Recall polynomial addition rule

Adding a constant \(b\) (which can be written as \(b + 0x+0x^{2}+\cdots+0x^{n}\)) to \(p(x)=a_{0}+a_{1}x + a_{2}x^{2}+\cdots+a_{n}x^{n}\) gives \((a_{0}+b)+a_{1}x + a_{2}x^{2}+\cdots+a_{n}x^{n}\), which is a polynomial.

Step2: Recall polynomial subtraction rule

Subtracting \(b\) from \(p(x)\) gives \((a_{0}-b)+a_{1}x + a_{2}x^{2}+\cdots+a_{n}x^{n}\), which is also a polynomial.

Answer:

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