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 + \dots + 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 a polynomial, and the difference is an integer.
• 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 an integer, and the difference is a polynomial.

Explanation:

Define the given expressions

We are given a polynomial \(p(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n\) and an integer \(b\). An integer \(b\) can be considered a constant polynomial of degree 0, which is a specific type of polynomial.

Analyze the sum of the polynomial and the integer

Using the Polynomial Classification knowledge point, we express the sum of \(p(x)\) and \(b\):

$$ p(x) + b = (a_0 + b) + a_1x + a_2x^2 + \dots + a_nx^n $$

Since \(a_0\) and \(b\) are constants, their sum \((a_0 + b)\) is also a constant. The resulting expression is still a polynomial.

Analyze the difference of the polynomial and the integer

Similarly, we express the difference of \(p(x)\) and \(b\):

$$ p(x) - b = (a_0 - b) + a_1x + a_2x^2 + \dots + a_nx^n $$

Since \((a_0 - b)\) is a constant, this expression is also a polynomial.

Evaluate the given options

• Option 1: "The sum of \(p(x)\) and \(b\) is a polynomial, and the difference is an integer." (Incorrect, the difference is a polynomial)
• Option 2: "The sum and difference of \(p(x)\) and \(b\) are integers." (Incorrect, they are polynomials)
• Option 3: "The sum and difference of \(p(x)\) and \(b\) are polynomials." (Correct, both operations result in polynomials)
• Option 4: "The sum of \(p(x)\) and \(b\) is an integer, and the difference is a polynomial." (Incorrect)

Answer:

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