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Chapter 5: Problem 186

Factor each polynomial completely. $$-2 z^{2}-16 z-32$$

Short Answer

Expert verified
-2(z + 4)^{2}

Step by step solution

01

Title - Factor out the greatest common factor (GCF)

First, identify the greatest common factor (GCF) of the polynomial. In this case, the GCF is \(-2\). Factor \(-2\) out of the polynomial: \(-2(z^{2} + 8z + 16)\)
02

Title - Recognize the quadratic trinomial

The expression inside the parentheses is a quadratic trinomial \(z^{2} + 8z + 16\). This trinomial can be factored further if it forms a perfect square trinomial.
03

Title - Factor the quadratic trinomial

Determine if the quadratic trinomial \(z^{2} + 8z + 16\) is a perfect square trinomial. A perfect square trinomial takes the form \(a^{2} + 2ab + b^{2} \). Here \(a = z\) and \((b = 4)\). Thus, \((z+4)^{2}\). Rewriting the polynomial, we get \(-2(z + 4)^{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

greatest common factor
In the world of polynomials, the greatest common factor (GCF) is a helpful concept to simplify expressions.
The GCF is the largest factor that divides each term of the polynomial without leaving a remainder.
Identifying the GCF helps in reducing the polynomial to simpler terms.
For the polynomial \(-2z^{2} - 16z - 32\), \(-2\) is the GCF. Factoring out the GCF is the first step:
  • \(-2z^{2} \, -2 \cdot \left(z^{2}\right) = -2z^{2}\)
  • \(-16z \, -2 \cdot \left(8z\right) = -16z\)
  • \(-32 \, -2 \cdot \left(16\right) = -32\)
Thus, the polynomial simplifies to \(-2(z^{2} + 8z + 16)\). Removing the GCF makes it easier to recognize and factor additional patterns in the polynomial.
quadratic trinomial
Next, we examine the quadratic trinomial inside the parentheses: \(z^{2} + 8z + 16\).
A quadratic trinomial is an expression of the form \(ax^{2} + bx + c\), where \(a\), \(b\), and \(c\) are constants.
The goal is to factor this expression into a product of two binomials or recognize it as a special type of trinomial.
For \(z^{2} + 8z + 16\):
  • The coefficient of \(z^{2}\) (a) is 1.
  • The coefficient of \(z\) (b) is 8.
  • The constant term (c) is 16.
Next, we check if it can be decomposed into simpler factors.
perfect square trinomial
Perfect square trinomials are a specific category of quadratic trinomials that follow a recognizable pattern.
They take the form \(a^{2} + 2ab + b^{2}\). To determine if \(z^{2} + 8z + 16\) is a perfect square trinomial, follow these steps:
  • Identify \(a\) and \(b\): Here, \(a = z\) and \((b = 4)\).
  • Check if \(2ab\) matches the middle term: \- \(2 \cdot z \cdot 4 = 8z\), which is the middle term in \(z^{2} + 8z + 16\).
  • Verify if the constant term equals \(b^{2}\): \- \(4^{2} = 16\).
Since all conditions match, \(z^{2} + 8z + 16\) is indeed a perfect square trinomial.
It can be factored as \((z + 4)^{2}\). Therefore, the original polynomial simplifies to \(-2(z + 4)^{2}\). Recognizing this pattern is very helpful in simplifying quadratic trinomials quickly.

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