Your task is to:

1. Formulate two real-life problems involving the factors of polynomials.
2. Solve these problems accurately and show the complete solution.

Here are two example problems and their solutions:

Problem 1:

Situation: You are planning to create a rectangular garden, and you know that the area of the garden is represented by the polynomial $A(x) = x^2 + 5x + 6$, where $x$ is the width in meters.

Task: Factor the polynomial to determine the possible dimensions of the garden.

Solution:

1. Factor the polynomial $x^2 + 5x + 6$. $A(x) = (x + 2)(x + 3)$
2. The factors give the possible dimensions of the garden. So, the width is $x + 2$ meters and the length is $x + 3$ meters.

Thus, the dimensions of the garden could be $x+2$ meters and $x+3$ meters.

Problem 2:

Situation: A box has a volume that can be represented by the polynomial $V(x) = x^3 - 7x^2 + 10x$, where $x$ represents the height of the box.

Task: Factor the polynomial to determine the possible dimensions of the box.

Solution:

1. Factor the polynomial $x^3 - 7x^2 + 10x$. $V(x) = x(x^2 - 7x + 10)$
2. Now, factor $x^2 - 7x + 10$ further: $V(x) = x(x - 5)(x - 2)$
3. The factors $x$, $x-5$, and $x-2$ represent the possible dimensions of the box. So, the height could be $x$, the width $x-5$, and the length $x-2$.

Thus, the possible dimensions of the box are $x$, $x-5$, and $x-2$.

Would you like more details or have any questions?

Here are 5 questions to expand on these concepts:

1. How do you determine if a polynomial can be factored easily?
2. Can every polynomial be factored, or are there exceptions?
3. What real-life scenarios might involve higher-degree polynomials?
4. How can factoring help in solving quadratic equations?
5. What are some common methods for factoring polynomials?

Tip: When factoring polynomials, always check if there’s a common factor across all terms first!