To help you formulate two real-life problems involving factors of polynomials and solve them step-by-step, here are two simple examples:

Problem 1: Fencing a Rectangular Garden

You have a rectangular garden, and you want to build a fence around it. The area of the garden is represented by the polynomial $A(x) = x^2 + 5x + 6$, where $x$ is the width of the garden in meters. Find the dimensions (length and width) of the garden by factoring the polynomial.

Step-by-Step Solution:

1. Area Polynomial: $A(x) = x^2 + 5x + 6$
2. Factor the polynomial: We are looking for two numbers that multiply to give $6$ (the constant term) and factor the quadratic equation as: $A(x) = (x + 2)(x + 3)$
3. Interpret the factors: The factors $(x + 2)$ and $(x + 3)$ represent the dimensions of the garden.
   • Width $= x + 2$ meters
   • Length $= x + 3$ meters

Conclusion:

The dimensions of the garden are $x + 2$ meters for width and $x + 3$ meters for length. If $x = 2$ meters, then the width is 4 meters and the length is 5 meters.

Problem 2: Designing a Flower Bed

A flower bed is in the shape of a rectangle. The area of the flower bed is given by $A(x) = 2x^2 + 7x + 3$, where $x$ represents the width in meters. Find the width and length of the flower bed by factoring the polynomial.

Step-by-Step Solution:

1. Area Polynomial: $A(x) = 2x^2 + 7x + 3$

2. Factor the polynomial: We factor the quadratic as follows:

   • Multiply $2 \times 3 = 6$.
   • We need two numbers that multiply to give $6$ and add to give $7$.
   • These numbers are $6$ and $1$.

   Now, we rewrite the middle term using these numbers: $A(x) = 2x^2 + 6x + x + 3$ Factor by grouping: $A(x) = 2x(x + 3) + 1(x + 3)$ Finally, factor out $(x + 3)$: $A(x) = (2x + 1)(x + 3)$

3. Interpret the factors: The factors $(2x + 1)$ and $(x + 3)$ represent the dimensions of the flower bed.

   • Width $= 2x + 1$ meters
   • Length $= x + 3$ meters

Conclusion:

The dimensions of the flower bed are $2x + 1$ meters for width and $x + 3$ meters for length. For example, if $x = 2$, then the width is $5$ meters, and the length is $5$ meters.

Would you like me to explain any of these steps further?

Relative Questions:

1. What are the methods for factoring quadratic polynomials?
2. How can you apply factoring in other real-life scenarios involving polynomials?
3. Can the quadratic formula be used as an alternative to factoring for these types of problems?
4. How would the problem change if the polynomial had a negative leading coefficient?
5. How do you check if a factorization is correct?

Tip:

When factoring polynomials, always look for common factors first, and remember that the signs in your binomials are important to ensure proper factorization.