Your task involves formulating two real-life problems related to factoring polynomials and solving them completely. Let me help you with both parts.

Problem 1: Area of a Rectangular Garden

A gardener wants to build a rectangular garden and knows that the total area of the garden will be given by the polynomial expression $x^2 + 5x + 6$, where $x$ represents some measurement of the garden's length in meters. The width of the garden is another factor of the polynomial.

Question: What are the possible dimensions of the garden?

Solution:

We factor the quadratic expression $x^2 + 5x + 6$:

$x^2 + 5x + 6 = (x + 2)(x + 3)$

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

Problem 2: Revenue Calculation for a Business

A company’s revenue in dollars is modeled by the polynomial expression $x^2 + 7x + 10$, where $x$ represents the number of units sold.

Question: What are the possible factors that contribute to the company’s revenue?

Solution:

We factor the polynomial $x^2 + 7x + 10$:

$x^2 + 7x + 10 = (x + 2)(x + 5)$

Thus, the factors contributing to the revenue are $x + 2$ and $x + 5$, which may represent the pricing model or number of units sold.

Would you like further details or have any specific questions?

Related Questions:

1. What is the importance of factoring polynomials in real-life applications?
2. How can we apply polynomial factoring in optimizing construction projects?
3. Can you formulate a similar problem related to physics or economics involving polynomials?
4. What are the key techniques to factor a quadratic polynomial quickly?
5. How does the distributive property relate to factoring polynomials?

Tip:

Always check if the quadratic expression can be factored by looking for factors of the constant term that add up to the middle coefficient.