Problem 93 - Algebra A Combined Function

Question

\(\left(x^{2}+6 x+9\right)-4 y^{2}\) (Hint: Factor the trinomial in parentheses first.)

Step-by-Step Solution

Answer

\((x + 3 - 2y)(x + 3 + 2y)\) is the factored expression.

Step 1: Identify the Trinomial

The expression inside the parentheses is \(x^2 + 6x + 9\). This is a quadratic trinomial.

Step 2: Factor the Trinomial

Recognize that \(x^2 + 6x + 9\) is a perfect square trinomial. It can be factored as \((x + 3)^2\) because \((x + 3)(x + 3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9\).

Step 3: Substitute the Factorization

Substitute the factored form back into the original expression: \((x + 3)^2 - 4y^2\).

Step 4: Recognize the Difference of Squares

The expression \((x + 3)^2 - 4y^2\) fits the form of a difference of squares, \(a^2 - b^2\), where \(a = (x + 3)\) and \(b = 2y\).

Step 5: Apply the Difference of Squares Formula

Use the formula \(a^2 - b^2 = (a - b)(a + b)\) to factor the expression: \(((x + 3) - 2y)((x + 3) + 2y)\).

Step 6: Simplify the Expression

The factored form is \((x + 3 - 2y)(x + 3 + 2y)\).

Key Concepts

Perfect Square Trinomial

Perfect square trinomials are a special type of quadratic expressions that can be rewritten as the square of a binomial. A perfect square trinomial follows the general form:

• \[ a^2 + 2ab + b^2 = (a + b)^2 \]

This means that the expression is the result of multiplying a binomial by itself.In the original exercise, the trinomial given is \(x^2 + 6x + 9\). Here, notice the coefficients: - The first term \(x^2\) is \(x\) squared. - The last term is \(9\), which is \(3^2\). - The middle term is \(6x\), which can be seen as \(2 \times x \times 3\).Thus, this expression is a perfect square trinomial, because it can be factored as \((x + 3)^2\). This simplification makes it easier to handle and leads to the understanding that perfect square trinomials help in seeing patterns.When analyzing expressions like this, it often saves time by looking for patterns that fit the perfect square form.

Difference of Squares

The difference of squares is another important algebraic structure that involves the subtraction of two squared terms. The general formula for factoring a difference of squares is:

• \[ a^2 - b^2 = (a - b)(a + b) \]

This formula emerges because when you multiply the two resulting binomials, the middle terms cancel out.For the original exercise, upon factoring the trinomial inside the parentheses, we get \((x + 3)^2 - 4y^2\). Let's see where this fits the difference of squares:

• \( a^2 = (x + 3)^2 \)
• \( b^2 = (2y)^2 \)

Using the difference of squares formula, you can rewrite the expression as \((x + 3 - 2y)(x + 3 + 2y)\).This type of factoring simplifies the expression into two binomials that make it easier to solve or use in further equations.

Quadratic Trinomial

Quadratic trinomials are three-term polynomial expressions in the form of \(ax^2 + bx + c\).When solving mathematical problems, recognizing a quadratic trinomial helps in deciding which factoring techniques to apply.In the context of the original exercise, \(x^2 + 6x + 9\) is identified as a quadratic trinomial because:

• The highest degree is two, with the term \(x^2\).
• There are two other terms à \(6x\) and \(9\).

The process involves recognizing patterns that allow the trinomial to be factored neatly in a way that simplifies the expression.Quadratic trinomials can often be expressed as a product of two binomials, through methods like factoring by grouping, applying the quadratic formula, or using simple observation based on perfect squares.In our exercise, it specifically leads to a perfect square trinomial which proves effective for further simplification when analyzing polynomial equations.