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Chapter 6: Problem 107

In the following exercises, factor completely using trial and error. $$ -16 x^{2}-32 x-16 $$

Short Answer

Expert verified
-16(x + 1)^2

Step by step solution

01

Identify the common factor

First, determine if there is a common factor in all terms. In the expression \( -16x^2 - 32x - 16 \), notice that each term has a common factor of \( -16 \).
02

Factor out the common factor

Factor out \( -16 \) from each term: \( -16(x^2 + 2x + 1) \).
03

Factor the quadratic expression

Recognize that the quadratic expression \( x^2 + 2x + 1 \) is a perfect square trinomial. It can be factored into \( (x + 1)(x + 1) = (x + 1)^2 \).
04

Write the final factored form

Combine the factors from the previous steps to write the completely factored form: \( -16(x + 1)^2 \).

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Key Concepts

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

Common Factor
When factoring polynomials, the first step is to look for any common factors. A common factor is a number or variable that divides all terms in an expression without leaving a remainder. For instance, in the expression \( -16x^2 - 32x - 16 \), we notice that each term is divisible by \( -16 \).
Factoring out the common factor simplifies the expression and makes it easier to work with. Here鈥檚 how we handle it. We divide each term by \( -16 \):
  • First term: \( -16x^2 / -16 = x^2 \)
  • Second term: \( -32x / -16 = 2x \)
  • Third term: \( -16 / -16 = 1\)
So, we get: \( -16(x^2 + 2x + 1) \).
This simpler expression is easier to tackle in the next steps of factoring.
Perfect Square Trinomial
A perfect square trinomial takes the form \( a^2 + 2ab + b^2 \) or \( a^2 - 2ab + b^2 \) and can be factored as \( (a + b)^2 \) or \( (a - b)^2 \). Recognizing these can simplify further factoring.
In our example, the quadratic expression inside the parentheses is \( x^2 + 2x + 1 \). We notice it matches the form \( a^2 + 2ab + b^2 \) where \( a = x \) and \( b = 1 \).
Therefore, \( x^2 + 2x + 1 \) can be factored as \( (x + 1)^2 \). This recognition helps conclude the factoring process efficiently.
Quadratic Expression
A quadratic expression is any polynomial of the form \( ax^2 + bx + c \), where a, b, and c are constants. These types of expressions are common in algebra, and learning how to factor them is crucial.
In our given problem, once we鈥檝e factored out the common \( -16 \), we are left with the quadratic expression \( x^2 + 2x + 1 \). Identifying the type of quadratic expression can guide us in choosing the right factoring method.
For example, recognizing this specific quadratic as a perfect square trinomial gives us a clear path to factor it further into \( (x + 1)(x + 1) \) or \( (x + 1)^2 \).
Thus, the fully factored form of the original polynomial is \(-16(x + 1)^2 \).

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Most popular questions from this chapter

Many trinomials of the form \(x^{2}+b x+c\) factor into the product of two binomials \((x+m)(x+n)\). Explain how you find the values of \(m\) and \(n\).

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