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Chapter 7: Problem 140

Factor completely. \(12 s^{2}+24 s+12\)

Short Answer

Expert verified
The factored form is \(12(s+1)^2\).

Step by step solution

01

Identify the greatest common factor (GCF)

To factor the given polynomial completely, start by identifying the greatest common factor (GCF) of the coefficients. The coefficients are 12, 24, and 12. The GCF of these numbers is 12.
02

Factor out the GCF

Next, factor out the GCF from each term in the polynomial:\[12s^2 + 24s + 12 = 12(s^2 + 2s + 1)\]
03

Factor the trinomial inside the parentheses

Now, focus on the quadratic trinomial inside the parentheses: \(s^2 + 2s + 1\). This trinomial is a perfect square. Notice that it can be factored as:\[(s + 1)^2\]Thus, we write it as:\[12(s + 1)^2\]

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Key Concepts

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

greatest common factor
To begin factoring a polynomial, you often start with finding the Greatest Common Factor (GCF). This is the largest number that can evenly divide all the coefficients in the polynomial. For the polynomial given in the exercise, the coefficients are 12, 24, and 12. These numbers can all be divided by 12, making 12 the GCF. By factoring out the GCF, you simplify the polynomial, making it easier to work with further.

For example, if you have the polynomial:
\(12s^2 + 24s + 12\)
Start by factoring out the 12:
\[12(s^2 + 2s + 1)\]
Factoring out the GCF ensures you're working with the simplest form of the polynomial, streamlining the rest of the factoring process.
quadratic trinomial
A quadratic trinomial is a polynomial of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The polynomial from the exercise simplifies to a quadratic trinomial once the GCF is factored out. From our example, we get:

\(s^2 + 2s + 1\)
A quadratic trinomial can often be factored further into the product of two binomials. In some cases, this trinomial might be a perfect square trinomial, which simplifies the factoring process. Identifying and factoring quadratic trinomials is a key step in solving polynomial equations, making the process quicker and more efficient.
perfect square
A perfect square trinomial has the form \(a^2 + 2ab + b^2\), and it can be factored into \((a + b)^2\). Recognizing a perfect square is helpful because it allows you to factor the expression quickly and accurately. In the exercise, the quadratic trinomial inside the parentheses, \(s^2 + 2s + 1\), is a perfect square. Notice that:

\((s + 1)^2 = s^2 + 2s + 1\)
Therefore, we can write our original polynomial as:
\[12(s + 1)^2\]
Factoring perfect squares involves identifying the pattern \(a^2 + 2ab + b^2\) and knowing that it matches the expanded form of \((a + b)^2\). This pattern simplifies the polynomial and provides a more compact and manageable form.

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