Problem 29 - Precalculus: Functions and Graphs

Question

Factor the polynomial. $$x^{2}+4 x+4-9 y^{2}$$

Step-by-Step Solution

Answer

The polynomial factors to \((x + 2 - 3y)(x + 2 + 3y)\).

Step 1: Arrange and group the expression

The given polynomial is \(x^2 + 4x + 4 - 9y^2\). Notice that it's composed of two main parts that suggest factoring: \(x^2 + 4x + 4\) and \(-9y^2\). Arrange it into a difference of squares form: \((x^2 + 4x + 4) - 9y^2\).

Step 2: Recognize the perfect square trinomial

The expression \(x^2 + 4x + 4\) is a perfect square trinomial. It can be factored into \((x + 2)^2\) because \((x + 2)(x + 2) = x^2 + 4x + 4\).

Step 3: Factor the difference of squares

Now the expression can be rewritten as \((x + 2)^2 - (3y)^2\). This is a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = (x + 2)\) and \(b = 3y\).

Step 4: Apply the difference of squares formula

Applying the formula gives us \([(x + 2) - 3y] [(x + 2) + 3y]\). Simplify this to \((x + 2 - 3y)(x + 2 + 3y)\).

Step 5: Write the final factored form

The completely factored form of the given polynomial is \((x + 2 - 3y)(x + 2 + 3y)\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that arises when two identical binomials are multiplied together. For example, when you expand

• urn (x + a)(x + a)
• you get the result (x + a)^2 = x^2 + 2ax + a^2.

This is a perfect square trinomial because the expression forms a perfect square.In the exercise you've seen, the expression \(x^2 + 4x + 4\) is recognized as a perfect square trinomial because it can be rewritten as \((x + 2)^2\). This is determined by identifying that both the leading and constant terms, \(x^2\) and 4, can be represented as squares:

• \(x^2 = x \times x\)
• \(4 = 2 \times 2\)
• And 2 times the product of \(x\) and \(2\) gives the middle term, \(2 \times x \times 2 = 4x\).

Recognizing such structures helps you simplify and factor the polynomial effectively.

Difference of Squares

The difference of squares is a fundamental concept in algebra that helps simplify expressions that look like one square subtracted from another. The general formula is

• \(a^2 - b^2 = (a - b)(a + b)\).

To recognize a difference of squares, the expression should contain two terms that are perfect squares, separated by a subtraction sign.For the exercise, after identifying

• \((x + 2)^2 - (3y)^2\)

we see that the term \((x + 2)^2\) is a perfect square of \((x + 2)\) and \(9y^2\) is a perfect square of \(3y\). Therefore, applying the difference of squares formula,

• we decompose the expression into \([(x + 2) - 3y][(x + 2) + 3y]\).

This factorization is useful as it breaks down complicated expressions into simpler products, which may be easier to solve or interpret in further mathematical operations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. These expressions can be manipulated and simplified using various algebraic rules and identities.In working with algebraic expressions, one of the key skills is recognizing patterns such as perfect square trinomials and differences of squares. These patterns help in factoring, which is the process of breaking down expressions into simpler multiples, making them easier to work with.Factoring

• simplifies algebraic expressions, aiding not just in solving equations, but also in finding roots or zeros of functions.
• It involves identifying common factors or special identities that make the expressions factorable into simpler components.

Understanding algebraic expressions and commands like the distributive property \((a + b)(c + d) = ac + ad + bc + bd\) and identities like \(a^2 - b^2 = (a - b)(a + b)\) is vital for anyone working through algebra problems. Being adept at recognizing different forms in expressions leads directly to faster and simpler problem-solving methods.