Problem 28 - Algebra and Trigonometry

Question

\(21-28\) Use a Factoring Formula to factor the expression. $$ 16 z^{2}-24 z+9 $$

Step-by-Step Solution

Answer

The factored expression is \((4z - 3)^2\).

Step 1: Identify Form

The expression given is \(16z^2 - 24z + 9\). This resembles the standard form of a perfect square trinomial, which is \(a^2 - 2ab + b^2 = (a - b)^2\). Here, our task is to determine if it can be expressed in this form.

Step 2: Find Values of 'a' and 'b'

Firstly, recognize that the first term \(16z^2\) is a perfect square, which can be written as \((4z)^2\). The third term \(9\) is also a perfect square, \((3)^2\). So, we identify \(a = 4z\) and \(b = 3\).

Step 3: Verify Middle Term

To verify if the trinomial is a perfect square, check if the middle term \(-24z\) is twice the product of \(a\) and \(b\). Calculate: \(-2ab = -2(4z)(3) = -24z\). This matches the middle term, confirming the trinomial is a perfect square.

Step 4: Write the Factored Form

Since the expression is confirmed to be a perfect square trinomial, we can write it in the factored form as \((a - b)^2\). Substitute the values of \(a\) and \(b\) to get \((4z - 3)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of polynomial expression that is formed by squaring a binomial. Essentially, it follows the pattern: \(a^2 - 2ab + b^2 = (a - b)^2\). Recognizing a perfect square trinomial is crucial for factoring because it allows you to simplify the expression into a more manageable form.

• The first and the last terms need to be perfect squares.
• The middle term should equal twice the product of the square roots of the first and last terms.

In the exercise we're analyzing, the expression \(16z^2 - 24z + 9\) demonstrates these properties. The first term, \(16z^2\), is a perfect square because it equals \((4z)^2\). Similarly, the last term, 9, equals \((3)^2\). These observations confirm that the expression can indeed be written as a squared binomial.

Factoring Formula

A factoring formula provides a structured method for breaking down polynomials into simpler expressions. For perfect square trinomials, the key formula is \((a - b)^2 = a^2 - 2ab + b^2\). This formula aids in transforming a seemingly complex polynomial into a simpler product of binomials.

In the given problem, identifying the appropriate \(a\) and \(b\) is the first step. For the expression \(16z^2 - 24z + 9\), we've concluded that \(a=4z\) and \(b=3\). Plugging these values into the factoring formula confirms our trinomial fits the \((a - b)^2\) pattern precisely.

• Step-by-step, this involves identifying the perfect squares.
• Ensuring the middle term is twice the product of the simpler components.

This method is invaluable in simplifying and then manipulating complex algebraic expressions into a more user-friendly format.

Polynomial Expressions

Polynomial expressions consist of variables and coefficients combined using operations of addition, subtraction, and multiplication. A polynomial can have terms with varying degrees, which refer to the variable's exponent in each term. Understanding how to handle polynomial expressions is a foundational skill in algebra.

• A term like \(16z^2\) is a polynomial term with a degree of 2.
• Expression \(16z^2 - 24z + 9\) is a trinomial as it has three terms.

Factoring polynomials is an essential process in algebra. It simplifies polynomials, making them easier to manage and solve in equations. Recognizing specific patterns, such as the perfect square trinomial, greatly enhances efficiency in reaching the correct form. Thus, polynomial manipulation is a crucial skill in diverse mathematical applications ranging from solving equations to integration in calculus.