Factor by any method. See Examples 1–7. 4z^2+28z+49

Verified step by step guidance

1. Step 1: Identify the quadratic expression. The given expression is \$4z^2 + 28z + 49$.
2. Step 2: Recognize that this is a perfect square trinomial. A perfect square trinomial takes the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\).
3. Step 3: Compare the given expression to the perfect square trinomial form. Here, \(a^2 = 4z^2\), \$2ab = 28z\(, and \)b^2 = 49$.
4. Step 4: Determine the values of \(a\) and \(b\). From \(a^2 = 4z^2\), we have \(a = 2z\). From \(b^2 = 49\), we have \(b = 7\).
5. Step 5: Verify that \$2ab = 28z\( holds true. Substitute \)a = 2z\( and \)b = 7\( into \)2ab\(: \)2(2z)(7) = 28z\(. This confirms the expression is a perfect square. Therefore, the factorization is \)(2z + 7)^2$.

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting a polynomial in the form ax^2 + bx + c as a product of two binomials. This process is essential for simplifying expressions and solving equations. For example, the expression 4z^2 + 28z + 49 can be factored to identify its roots and analyze its graph.

Perfect Square Trinomials

A perfect square trinomial is a specific type of quadratic expression that can be expressed as the square of a binomial. The general form is (a + b)^2 = a^2 + 2ab + b^2. In the case of 4z^2 + 28z + 49, recognizing it as a perfect square trinomial allows us to factor it as (2z + 7)^2.

The Distributive Property

The distributive property states that a(b + c) = ab + ac, which is fundamental in both expanding and factoring expressions. When factoring, we often reverse this property to find common factors or to express a polynomial as a product of simpler expressions. Understanding this property is crucial for manipulating algebraic expressions effectively.