Problem 80 - A Graphical Approach to College Algebra

Question

Factor by any method. $$q^{2}+6 q+9-p^{2}$$

Step-by-Step Solution

Answer

The expression factors to \((q+3-p)(q+3+p)\).

Step 1: Identify the Expression Structure

The given expression is \(q^{2} + 6q + 9 - p^{2}\). This is a polynomial expression with four terms, where the first three terms \(q^{2} + 6q + 9\) form a perfect square trinomial.

Step 2: Recognize Perfect Square Trinomial

Notice that \(q^{2} + 6q + 9\) is a perfect square trinomial. It can be expressed as \((q+3)^2\) because \((q+3)(q+3) = q^2 + 6q + 9\).

Step 3: Rewrite the Expression

Rewrite the given expression \(q^{2} + 6q + 9 - p^{2}\) using the perfect square trinomial, so it becomes \((q+3)^2 - p^2\).

Step 4: Identify the Difference of Squares

The expression \((q+3)^2 - p^2\) is a difference of squares. This can be factored using the identity \(a^2 - b^2 = (a-b)(a+b)\).

Step 5: Apply the Difference of Squares Formula

Apply the formula with \(a = (q+3)\) and \(b = p\), we get \((q+3)^2 - p^2 = ((q+3) - p)((q+3) + p)\).

Step 6: Simplify the Expression

Simplify the factored expression to \((q+3-p)(q+3+p)\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that is formed by squaring a binomial. This means that the trinomial can be written in the form

• \(a^2 + 2ab + b^2 = (a+b)^2\)

Here, the trinomial is created by squaring the sum of two terms.
In our original expression, we observe

• \(q^2 + 6q + 9\)

This fits the pattern of a perfect square trinomial where:

• \(q^2\) is the square of the first term
• \(9\) is the square of the last term
• \(6q\) is twice the product of the square roots of the first and last terms

Thus, it can be factored as

• \((q+3)^2\)

Recognizing this pattern is crucial for simplifying expressions and solving polynomial equations.

Difference of Squares

The difference of squares is another useful pattern in polynomial expressions. It is defined by the identity:

• \(a^2 - b^2 = (a-b)(a+b)\)

This format indicates that if you have a binomial consisting of two squared terms with a subtraction operator in between, you can factor it into two linear binomials.
In the expression

• \((q+3)^2 - p^2\)

we have a situation where

• \((q+3)^2\)

and

• \(p^2\)

are both perfect squares. This can be manipulated using the difference of squares identity to become:

• \((q+3-p)(q+3+p)\)

This technique is immensely powerful in algebra, providing a direct method to factor and simplify polynomial expressions.

Polynomial Factoring Techniques

Factoring is an essential algebraic skill that involves expressing a polynomial as a product of its simpler components. There are several techniques used in the factoring process, each applicable under different circumstances.
Key techniques include:

• Perfect Square Trinomials: When only one term can be concluded as a square of another term after expanding it. Example: \((a+b)^2\)
• Difference of Squares: As discussed, this is applicable when you encounter an expression of the form \(a^2 - b^2\).
• Greatest Common Factor (GCF): First, it’s crucial to check for any common factors that can be factored out from all the terms in the polynomial.
• Grouping: Useful when a polynomial has four terms. A common technique is to group terms in pairs, factor the GCF from each pair, and look for a common binomial factor.

Factoring breaks down complex expressions into simpler, solvable forms and lays the groundwork for solving polynomial equations, simplifying complex problems, and achieving polynomial division.