Problem 26 - Beginning and Intermediate Algebra

Question

Factor completely. $$9 j^{2}-18 j+9$$

Step-by-Step Solution

Answer

The completely factored form of the given expression \(9j^2 - 18j + 9\) is \(9(j - 1)^2\).

Step 1: Identify the common factor

First, we need to find a common factor among all the terms in the expression. In this case, the common factor is 9.

Step 2: Factor out the common factor

Next, we factor out the common factor from each term of the quadratic expression: \(9j^2 - 18j + 9 = 9(j^2 - 2j + 1)\)

Step 3: Factorize the quadratic expression inside the parenthesis

Now, we need to factorize the quadratic expression inside the parenthesis: \(j^2 - 2j + 1\). We can use factoring techniques such as finding roots or factoring by grouping to do this. In this case, the expression is already a perfect square trinomial because it's in the form \((j-A)^2\). We can factorize it as: \(j^2 - 2j + 1 = (j - 1)^2\)

Step 4: Combine the factored terms

Finally, integrate the factored terms from step 2 and step 3: \(9j^2 - 18j + 9 = 9(j - 1)^2\)

Key Concepts

Quadratic Expressions

A quadratic expression is a type of polynomial that has a degree of 2. This means the highest power of the variable is 2. A general example of a quadratic expression is like the equation \[ ax^2 + bx + c \]where

• \(a\), \(b\), and \(c\) are constants,
• \(a eq 0\) because if \(a\) is zero, then the expression wouldn't be quadratic anymore.

Quadratic expressions can usually be factored into a product of two binomials, making them simpler to solve or analyze. Recognizing the structure, like finding out if it's a perfect square trinomial, often helps in its simplification.

Common Factor

A common factor in a polynomial expression is a number or variable that divides each term without a remainder. To find a common factor, we look at the coefficients of each term and see if there is a largest number that works for all of them.
In the given example \[9j^2 - 18j + 9\], the greatest common factor is 9, as this is the largest number that can divide every term in the expression. By identifying the common factor first, we can often simplify the expression initially which makes further factoring much easier.
Once factored out, it reduces the complexity of the succeeding polynomial, in this case rendering \[j^2 - 2j + 1\] needing to be worked with further.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. This form is useful because it simplifies the factoring process, often recognized by a particular pattern. A perfect square trinomial is generally in a form like: \[(a - b)^2 = a^2 - 2ab + b^2.\]When comparing our example, \(j^2 - 2j + 1\), it fits the pattern of a perfect square trinomial, because it can be rewritten as \[(j - 1)^2.\]
Identifying a perfect square trinomial is valuable as it tells us immediately how the expression can be expressed as repeated roots. This means fewer steps are needed in the factoring process and offers a quick path to understanding the structure of the expression.