Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Completely factor the expression.\(9 x^{2}-6 x+1\)

Step-by-Step Solution

Answer

The factored form of the given quadratic expression \(9x^{2}-6x+1\) is \((3x - 1)^2\).

Step 1: Identify the coefficients and constant

The exercise gives the quadratic equation \(9x^{2}-6x+1\). Here, a=9, b=-6, and c=1.

Step 2: Find two numbers with specific sum and product

Find two numbers such that their product equals the multiplication of a and c (i.e., 9*1 = 9) and their sum equals the value of b which is -6. The numbers that satisfy these conditions are -3 and -3, because (-3)*(-3) = 9 and -3 + -3 = -6

Step 3: Factor the quadratic expression

Now, we can write the quadratic equation in the factored form. The equation \(9x^{2}-6x+1\) can be written as \((3x - 1)^2\) or (3x-1)(3x-1).

Key Concepts

Quadratic Equations

Quadratic equations are a central topic in algebra that often involve finding the values of x that satisfy a quadratic expression set to zero. These equations are usually written in the standard form of \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are coefficients and \(a \eq 0\). In our exercise, the expression to factor is \(9x^2-6x+1\), which could be equated to zero to find the roots of the corresponding quadratic equation.

Factoring is the process used to rewrite a quadratic expression as a product of its binomials. When the quadratic is a perfect square trinomial, like in our example, it can be factored into the square of a binomial (\(3x-1\))^2. Understanding how to factor quadratic expressions is crucial for solving quadratic equations and is often a stepping stone to more complex algebraic operations.

Factoring Polynomials

The process of factoring polynomials requires identifying expressions that can multiply together to give the original polynomial. It's like breaking down a complex structure into its building blocks. For a quadratic expression, this usually involves finding two binomials that multiply to provide the original trinomial.

In our step-by-step solution, we applied the understanding that for a polynomial \(ax^2 + bx + c\), we can seek two numbers whose sum is \(b\) and whose product is \(ac\). This process can significantly simplify solving equations and understanding the graphs and properties of quadratic functions. In more complex cases, factoring may involve techniques such as grouping or utilizing the quadratic formula when the expression does not factor neatly.

Solving Algebraic Expressions

Solving algebraic expressions generally involves simplifying or rearranging the equation to find the value of the unknown variable. Factoring is a powerful tool for simplifying these expressions, particularly when combined with other algebraic techniques. Once an expression is factored, as seen with our expression \(9x^2-6x+1\), solving for x becomes much more manageable.

Factored forms are beneficial because they highlight the roots of the equation—the values of x for which the expression equals zero. For our particular expression, the roots would be the values of x that make \(3x-1=0\), which further simplifies the solving process. Mastery of factoring and understanding its relationship to the zeros of functions opens doors to deeper exploration of algebraic concepts, such as graph analysis and the behavior of functions.