Question

A Rectangular Field with an area of r^2 + 9x + 18 square meters represents the area of a rectangular field. Find two binomials that might represent the length and width of the field.

a) (r + 3)(r + 6)
b) (r + 3)(r + 2)
c) (r + 9)(x + 2)
d) (r^2 + 9)(x + 18)

Answer 1

The answer of the given equation that "A Rectangular Field with an area of r^2 + 9x + 18 square meters represents the area of a rectangular field. Find two binomials that might represent the length and width of the field.

" is a)
`\((r + 3)(r + 6)\)`

Explanation:

The area of a rectangular field is given by the product of its length and width. In this case, the area is represented by the expression
`\(r^2 + 9x + 18\).`

To find two binomials that might represent the length and width of the field, we look for two expressions whose product gives the given area.

The correct binomials are found by factoring the given expression. The factorization of
`\(r^2 + 9x + 18\) is \((r + 3)(r + 6)\).`

Therefore, the correct answer is option a)
`\((r + 3)(r + 6)\).`

Answer 2

To factorize the expression r^2 + 9x + 18 into two binomials, finding two numbers that multiply to 18 and add up to 9 leads to 3 and 6. Hence, the binomials (r + 3) and (r + 6) represent the length and width of the field, resulting in (r + 3)(r + 6) when multiplied. Therefore the correct option is a.

Step-by-step explanation:

The given expression for the area of the rectangular field is r^2 + 9x + 18 square meters. To factorize this expression into two binomials representing the length and width of the field, we need to find two numbers that multiply to give 18 and add up to give 9. These numbers are 3 and 6 since 3 * 6 = 18 and 3 + 6 = 9.

Therefore, the two binomials that represent the length and width of the field are (r + 3) and (r + 6), resulting in the expression (r + 3)(r + 6) when multiplied together using the distributive property.

Factorization plays a crucial role in simplifying expressions, solving equations, and understanding the properties of numbers and algebraic structures. It is fundamental in various mathematical concepts and has applications across different fields, including algebra, number theory, and cryptography. Therefore the correct option is a.