Question

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 $4x^2 + 9$.

image of a rectangle labeled patio with width x and length 4x, and a square labeled garden with side length 3 next to it

Explanation:

Question 8

• Closure under an operation means that when we perform the operation on two elements of a set (here, polynomials), the result is also an element of that set.
• Addition: When adding two polynomials, we combine like terms. The sum of two polynomials is always a polynomial. For example, \((2x + 3)+(4x^2 - 2x + 1)=4x^2 + 4\), which is a polynomial.
• Subtraction: Subtracting two polynomials is similar to adding (with a sign change). For example, \((4x^2 + 3)-(2x^2 + x - 1)=2x^2 - x + 4\), which is a polynomial.
• Multiplication: When multiplying two polynomials, we use the distributive property (FOIL for binomials, extended for higher degrees). The product of two polynomials is a polynomial. For example, \((x + 2)(x - 3)=x^2 - 3x + 2x - 6=x^2 - x - 6\), a polynomial.
• Division: Polynomials are not closed under division. For example, dividing the polynomial \(x\) by the polynomial \(x^2\) gives \(\frac{1}{x}\), which is not a polynomial (it has a negative exponent, violating the definition of a polynomial).

So, polynomials are closed under addition, subtraction, and multiplication.

Answer:

A. Addition, B. Subtraction, C. Multiplication