Question

1. amanda says the gcf of $8m^2n$ and $4mn$ is 4. kris says the gcf is $4mn$. which student is correct? explain your answer.17. the area of a rectangular table is given by the trinomial $w^2 - 11w + 24$. what are the possible dimensions of the table?18. the area of a rectangular football field is $n^2 - 6n - 40$. what are the possible dimensions of the football field?

Explanation:

Question 16

Step1: Factor each term

Factor \(8m^{2}n\): \(8m^{2}n = 2\times2\times2\times m\times m\times n\)
Factor \(4mn\): \(4mn = 2\times2\times m\times n\)

Step2: Identify GCF

The common factors are \(2\times2\times m\times n=4mn\). Amanda only considered the numerical GCF, but GCF of monomials includes common variables with lowest exponents. So Kris is correct.

Step1: Factor the trinomial

We need to factor \(w^{2}-11w + 24\). Find two numbers that multiply to \(24\) and add to \(- 11\). The numbers are \(-3\) and \(-8\) (since \((-3)\times(-8)=24\) and \(-3+(-8)=-11\)).
So \(w^{2}-11w + 24=(w - 3)(w - 8)\)

Step2: Determine dimensions

For a rectangle, area = length × width. So the possible dimensions are \((w - 3)\) and \((w - 8)\) (assuming \(w>8\) for positive dimensions).

Step1: Factor the trinomial

Factor \(n^{2}-6n - 40\). Find two numbers that multiply to \(-40\) and add to \(-6\). The numbers are \(4\) and \(-10\) (since \(4\times(-10)=-40\) and \(4+(-10)=-6\)).
So \(n^{2}-6n - 40=(n + 4)(n - 10)\)

Step2: Determine dimensions

For a rectangle, area = length × width. So the possible dimensions are \((n + 4)\) and \((n - 10)\) (assuming \(n>10\) for positive dimensions).

Answer:

Kris is correct. The GCF of \(8m^{2}n\) and \(4mn\) is \(4mn\) (includes both numerical and variable factors), while Amanda only considered the numerical part.

Question 17