Problem 75 - A Graphical Approach to College Algebra

Question

Factor by any method. $$4 z^{2}+28 z+49$$

Step-by-Step Solution

Answer

The expression \(4z^2 + 28z + 49\) factors as \((2z + 7)^2\).

Step 1: Identify the Form

The given expression is a quadratic trinomial: \(4z^{2} + 28z + 49\). We want to determine if it can be written in the form \((az + b)^2\). This expression resembles a perfect square trinomial.

Step 2: Check the Coefficients

In a perfect square trinomial \((az + b)^2 = a^2z^2 + 2abz + b^2\), the first and last terms should be perfect squares. Here,- First term: \(4z^2 = (2z)^2\)- Last term: \(49 = 7^2\)

Step 3: Verify Middle Term

The middle term in a perfect square trinomial should be \(2abz\), where \(a=2\) and \(b=7\). Calculate:\(2 imes 2 imes 7 = 28\), which matches the middle term of the original expression \(28z\).

Step 4: Write the Expression as a Square

Since all the terms align with the perfect square trinomial pattern, we can write the expression as:\((2z + 7)^2\).

Step 5: Factor the Quadratic

The factorization of \(4z^2 + 28z + 49\) is \((2z + 7)(2z + 7)\) or simply \((2z + 7)^2\).

Key Concepts

Perfect Square Trinomial

One powerful technique in factoring quadratics is recognizing a perfect square trinomial, a special type of quadratic expression. A perfect square trinomial is one that can be expressed as the square of a binomial. The general form is \((ax + b)^2 = a^2x^2 + 2abx + b^2\).
For example, consider the trinomial \(4z^2 + 28z + 49\). To determine if it's a perfect square trinomial, identify if both the first and last terms are perfect squares:

• The first term \(4z^2\) is \((2z)^2\).
• The last term \(49\) is \(7^2\).

Next, check the middle term. It should be twice the product of the numbers taken in the square roots of the first and last terms, i.e., \(2 \cdot 2z \cdot 7\), which indeed equals \(28z\), confirming it as a perfect square trinomial. Finally, write the expression as \((2z + 7)^2\).
Recognizing these patterns makes factoring quicker and simpler, transforming a quadratic into an easily manageable form.

Quadratic Trinomials

Quadratic trinomials are algebraic expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Understanding these forms is crucial as they appear widely in algebraic problems.
In the exercise, \(4z^2 + 28z + 49\), \(a = 4\), \(b = 28\), and \(c = 49\).
Quadratic trinomials appear in various forms; finding effective methods to factor them can simplify problem-solving. Methods include:

• Finding common factors and factoring them out,
• Recognizing patterns like perfect square trinomials,
• Using the quadratic formula when applicable.

Identifying these patterns, like those in a perfect square trinomial, facilitates breaking down the quadratic trinomial effectively. The resulting product of factors is often easier to interpret or use in further algebraic manipulations.

Factoring Techniques

Factoring is a strategy used to simplify expressions, solve equations, and scrutinize expression structures. It is particularly useful when working with quadratic equations, allowing for easy solutions.
In our example, the trinomial \(4z^2 + 28z + 49\) is factored by recognizing it as a perfect square trinomial. Factoring techniques vary based on the expression's form.

• Recognize perfect square trinomials, as we did: \((2z + 7)^2\).
• Factor by grouping, which can simplify more complex expressions into two binomials.
• Employ the Distributive Property reversely to write expressions as a product.

These techniques can solve different algebraic problems flexibly and efficiently. Mastering them enables easier manipulation of equations, providing a foundational skill in more advanced algebra. With practice, identifying which technique to apply becomes intuitive.