Problem 76 - Elementary and Intermediate Algebra

Question

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ z^{2}+6 y z^{2}+9 y^{2} z^{2} $$

Step-by-Step Solution

Answer

The factored form is \( z^{2}(1 + 3y)^2 \).

Step 1: Identify the Expression

The given expression is a quadratic trinomial in terms of \(z\) with coefficients that involve \(y\): \[ z^{2}+6 y z^{2}+9 y^{2} z^{2} \]

Step 2: Rewrite for Common Factoring

Observe that each term in the expression has \(z^{2}\) as a factor. Factor \(z^{2}\) out from each term, rewriting the expression as: \[ z^{2}(1 + 6y + 9y^2) \]

Step 3: Recognize a Perfect Square Trinomial

The expression inside the parenthesis, \(1 + 6y + 9y^2\), is a perfect square trinomial. This can be rewritten as \((1 + 3y)^2\) because it follows the form \(a^2 + 2ab + b^2\) with \(a = 1\) and \(b = 3y\).

Step 4: Write the Final Factored Form

Insert the factorization of the perfect square trinomial back into the expression. The fully factored form of the original expression is: \[ z^{2}(1 + 3y)^2 \]

Key Concepts

Perfect Square Trinomial

In algebra, recognizing and working with perfect square trinomials simplifies factoring significantly. A perfect square trinomial is a special form of a quadratic expression. It can be identified by its form:

• \( a^2 + 2ab + b^2 \)

You can factor this form into the square of a binomial. The expression becomes:

• \( (a + b)^2 \)

A perfect square trinomial allows you to easily simplify and solve expressions, particularly when these appear inside more complex polynomials. For instance, in the quadratic expression \(1 + 6y + 9y^2\), if we identify it as a perfect square by setting \(a = 1\) and \(b = 3y\), we see it matches \(a^2 = 1\), \(2ab = 6y\), and \(b^2 = 9y^2\).
Thus, we can transform \(1 + 6y + 9y^2\) into \((1 + 3y)^2\). This neat little trick allows for simpler expressions and easier calculations later on.

Factoring Techniques

Factoring techniques are crucial skills in algebra that help simplify expressions and solve equations. These techniques involve breaking down complex expressions into simpler, more manageable pieces or factors.
Factoring can take many forms, such as:

• Extracting common factors, as was done with \(z^2\) in the original exercise.
• Utilizing identities like difference of squares or sum of cubes.
• Recognizing special cases such as perfect square trinomials.

In our example, the initial step was to find the greatest common factor, which was \(z^2\), and factor it out. This left us with the simpler trinomial \(1 + 6y + 9y^2\), which could then be easily recognized and factored as a perfect square trinomial.
The key is to recognize these patterns and apply the appropriate method, simplifying the process and revealing solutions more clearly.

Algebraic Identities

Algebraic identities are formulas or equations that hold true for all values of the variables involved. They are fundamental in simplifying expressions and solving equations. These identities serve as shortcuts that capture relationships between variables, helping to recognize patterns and apply them effectively.
Some important algebraic identities that assist in factoring include:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)
• \(a^2 - b^2 = (a + b)(a - b)\)

Using these identities in our problem, we identified \(1 + 6y + 9y^2\) as \((1 + 3y)^2\), utilizing the perfect square trinomial identity. This powerful tool facilitated the further simplification of the expression, transforming it into a product of factors that is much easier to manage in equations or further analysis. Being familiar with algebraic identities allows for quicker and more accurate solutions in mathematics.