Question

a. closure
b. commutative
c. addition property of equality
d. zero identity

1. under which operations are polynomials closed?

a. addition
b. subtraction
c. multiplication
d. division

use this scenario to answer questions 9 – 10.

the area of a rectangular patio is $4x^2$. a square garden is built next to the patio. the total area of the patio and the garden is $4x^2 + 9$.

Explanation:

Question 8

To determine under which operations polynomials are closed, we analyze each operation:

• Addition: When adding two polynomials, we combine like terms. The sum of two polynomials (e.g., \( (a_nx^n+\dots+a_1x + a_0)+(b_nx^n+\dots+b_1x + b_0)=(a_n + b_n)x^n+\dots+(a_1 + b_1)x+(a_0 + b_0) \)) is also a polynomial, so polynomials are closed under addition.
• Subtraction: Subtracting two polynomials (\( (a_nx^n+\dots+a_1x + a_0)-(b_nx^n+\dots+b_1x + b_0)=(a_n - b_n)x^n+\dots+(a_1 - b_1)x+(a_0 - b_0) \)) results in a polynomial, so they are closed under subtraction.
• Multiplication: Multiplying two polynomials (using the distributive property) results in a sum of terms with non - negative integer exponents and coefficients, which is a polynomial. For example, \( (ax + b)(cx + d)=acx^2+(ad + bc)x+bd \).
• Division: If we divide two polynomials, we may get a non - polynomial (e.g., dividing \( x \) by \( x^2 \) gives \( \frac{1}{x} \), which is not a polynomial). So polynomials are not closed under division.

So the operations under which polynomials are closed are addition, subtraction, and multiplication. The options A (Addition), B (Subtraction), and C (Multiplication) are correct.

Answer:

A. Addition, B. Subtraction, C. Multiplication