Problem 110 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Suppose a square garden has an area represented by \(9 x^{2}\) square feet. If one side is made 7 feet longer and the other side is made 2 feet shorter, then the trinomial that models the area of the larger garden is \(9 x^{2}+15 x-14\) square feet.

Step-by-Step Solution

Answer

The statement is false. The correct trinomial that models the area of the larger garden is \(9 x^{2}+ x -14\) square feet.

Step 1: Calculate the new area

Calculate the new area of the garden using the new dimensions. Multiply \(3x+7\) by \(3x-2\) to get \(9x^2+x-14\).

Step 2: Compare areas

Compare \(9x^2+x-14\) (new area) with the given area in the problem, \(9x^2+15x-14\). Since they are not equal (the middle term differs), the statement is false.

Step 3: Correct the statement

To correct the statement for it to be true, replace the incorrect expression with the correct expression for the larger garden area. The correct statement should read: Suppose a square garden has an area represented by \(9 x^{2}\) square feet. If one side is made 7 feet longer and the other side is made 2 feet shorter, then the trinomial that models the area of the larger garden is \(9 x^{2}+ x -14\) square feet.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like x or y), and operation signs (such as +, -, *, /). They are essential in representing real-world problems in mathematical language, allowing us to work out solutions systematically. For instance, the expression for the area of the original square garden in our problem, 9 x^{2}, indicates the area is nine times the square of the length of the garden's side.

Understanding how to manipulate these expressions is fundamental in various areas of mathematics, including solving polynomial area problems. In the case of our garden, modifying the garden’s dimensions leads to a change in the algebraic expression representing the area. As such, a well-grasped knowledge on how to work with algebraic expressions will be useful in correcting the false statement provided in the exercise.

Factoring Polynomials

Factoring polynomials is the process of breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. In our exercise, we are dealing with a trinomial, which is a polynomial with three terms. Understanding the rules of factoring can help in both simplifying expressions and solving equations.

An essential application of factoring arises when correcting the false statement given in the exercise. By expressing the sides of the new garden as (3x + 7) and (3x - 2) and factoring the new area, we find the correct trinomial is 9x^2 + x - 14. This highlights the importance of factoring skills when trying to find the correct algebraic expression representing a specific geometrical figure such as our garden.

Solving Quadratic Equations

Solving quadratic equations is a critical skill in algebra that involves finding the values of the variable that make the equation true. These equations are of the form ax^2 + bx + c = 0. Though it might seem tangential to our exercise at first glance, understanding how to solve quadratic equations can deeply enhance one's ability to tackle polynomial area problems.

When a quadratic equation is set equal to a value (like the area of a geometrical figure), finding solutions to the equation can provide insights about the dimensions of that figure. In the context of our exercise, if we had to solve for the side lengths after knowing the area, we would set up a quadratic equation and solve for x. This area of knowledge, while not directly applied in the step-by-step solution provided, is an underlying skill that provides a richer understanding and mastery over these kind of mathematical challenges.