Problem 72 - 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. $$ 16 d^{2}-56 d z+49 z^{2} $$

Step-by-Step Solution

Answer

The expression is \((4d - 7z)^2\).

Step 1: Identify the Structure

Look at the given expression \(16d^2 - 56dz + 49z^2\). Notice that it has the form of a quadratic trinomial which can resemble the square of a binomial.

Step 2: Verify Perfect Square Trinomial

Check if each term is a perfect square. \(16d^2\) is \((4d)^2\), \(49z^2\) is \((7z)^2\). The middle term can be checked using \(-2\times (4d)\times (7z) = -56dz\), which matches.

Step 3: Write as a Binomial Square

Since the expression fits the pattern \(a^2 - 2ab + b^2 = (a-b)^2\), where \(a = 4d\) and \(b = 7z\), factor the expression as \((4d - 7z)^2\).

Key Concepts

Understanding the Binomial Square

A binomial square involves multiplying a binomial by itself. Let's consider the binomial \[(a + b)^2\]\. The result is a quadratic expression \[a^2 + 2ab + b^2\]. In this pattern, the expressions are expanded using:

• \(a^2\) - the square of the first term
• \(2ab\) - twice the product of the two terms
• \(b^2\) - the square of the second term

Understanding this format helps in recognizing expressions that can be rewritten as squares of binomials. For example, take \[(4d - 7z)^2\]. When expanded, this expression becomes \[16d^2 - 56dz + 49z^2\]. Thus, recognizing this structure allows easy transformation from an expanded quadratic form back to a binomial square.

What is a Perfect Square Trinomial?

A perfect square trinomial is a specific type of quadratic trinomial. It takes the form \[a^2 + 2ab + b^2\]. This trinomial originates from squaring a binomial. To identify a perfect square trinomial, check:

• First and last terms should be perfect squares, e.g., \(16d^2\) and \(49z^2\).
• The middle term should be twice the product of the square roots of the first and last terms, \(2\times 4d \times 7z = 56dz\).

If these conditions hold, you can confidently state the trinomial is a perfect square trinomial.Recognizing perfect square trinomials enables you to factor expressions easily, as demonstrated in transforming, \[16d^2 - 56dz + 49z^2\] into \[(4d - 7z)^2\]. This transformation is beneficial in simplifying complex expressions.

The Role of Quadratic Expressions

Quadratic expressions are foundational in algebra and take the standard form: \[ax^2 + bx + c\]. These expressions contain variables raised to the second power. They appear in various math problems from geometry to physics.Quadratic trinomials, like \[16d^2 - 56dz + 49z^2\], are among the types of quadratic expressions.They result from combining squares and their products. Factoring them, specifically when they become perfect square trinomials, simplifies the quadratic into a compact binomial form.Solving or simplifying quadratic expressions often involves recognizing patterns that allow you to rewrite them neatly, like turning trinomials into binomial squares. This restructuring is particularly useful for solving quadratic equations or optimizing functions in calculus. Understanding these forms makes solving complex mathematical problems more approachable and less daunting.