Problem 273 - Elementary Algebra

Question

Sue and Alan are planning to put a 15 foot square swimming pool in their backyard. They will surround the pool with a tiled deck, the same width on all sides. If the width of the deck is \(w\), the total area of the pool and deck is given by the trinomial \(4 w^{2}+60 w+225\). Factor the trinomial.

Step-by-Step Solution

Answer

The trinomial factors to \[ (2w + 15)^2 \].

Step 1: Recognize the trinomial

The given trinomial is \[ 4w^2 + 60w + 225 \]. Notice that this represents the total area of the pool and the deck.

Step 2: Find the square terms

Identify the square terms. In this trinomial, the first term is \[ 4w^2 \] and the last term is \[ 225 \].

Step 3: Write the trinomial as a perfect square

Recognize the form of \[ (ax + b)^2 \]. Here, think of how \[ (2w + 15)^2 \] expands: \[ (2w)^2 + 2 \times 2w \times 15 + 15^2 = 4w^2 + 60w + 225 \].

Step 4: Confirm the factor

Verify that \[ (2w + 15)^2 \] indeed expands to the given trinomial: \[ (2w + 15)(2w + 15) = 4w^2 + 60w + 225 \].

Step 5: State the factored form

Thus, the factored form of the trinomial \[ 4w^2 + 60w + 225 \] is \[ (2w + 15)^2 \].

Key Concepts

Trinomial Factorization

Trinomial factorization is the process of breaking down a polynomial with three terms, called a trinomial, into simpler components called factors. In this exercise, we are given the trinomial \(4w^2 + 60w + 225\). To factor a trinomial effectively, follow these steps:

• Identify the three terms.
• Look for common factors in each term.
• Rewrite the trinomial in a recognizable squared format, if possible.

For example, let's consider the given trinomial: \(4w^2 + 60w + 225\). Notice that this can be rewritten as \( (2w + 15)^2 \) because it fits the pattern of a perfect square trinomial (which we'll discuss in the next section). By rewriting the original trinomial in this squared format, we have effectively factored it.

Perfect Square Trinomials

Perfect square trinomials are special types of trinomials that can be expressed as the square of a binomial. They take the form \( (ax + b)^2 \). This translates into \(a^2x^2 + 2abx + b^2.\)

• The first term \(a^2x^2\) is a square term.
• The last term \(b^2\) is a square term.
• The middle term \(2abx\) is twice the product of the square roots of the first and last terms.

Using the given trinomial \(4w^2 + 60w + 225\), we can identify:

• The first term \(4w^2\) is the square of \(2w\).
• The last term \(225\) is the square of \(15\).
• The middle term \(60w\) is twice the product of \(2w\) and \(15\). This confirms that \(4w^2 + 60w + 225\) is indeed a perfect square trinomial.

Thus, we can factor it as \((2w + 15)^2\).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition and multiplication). They form the building blocks of algebra and are essential for solving problems like the one presented in this exercise.

In the context of our example, \(4w^2 + 60w + 225\), we have a quadratic trinomial—an algebraic expression involving a variable \(w\). To handle algebraic expressions:

• Understand the meaning of each term.
• Use properties of operations like distribution and factoring.
• Recognize patterns such as perfect square trinomials to simplify expressions.

By applying these concepts, we transformed the expression \(4w^2 + 60w + 225\) into its factored form \((2w + 15)^2\), allowing us to solve or analyze the problem more effectively.