Problem 78 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(x^{2}-6 x+9\)

Step-by-Step Solution

Answer

The expression factors to \((x - 3)^2\).

Step 1: Identify the Structure of the Expression

The expression given is a quadratic trinomial of the form \(ax^2 + bx + c\), where \(a = 1\), \(b = -6\), and \(c = 9\). To factor, we look for two numbers that multiply to \(c = 9\) and add to \(b = -6\).

Step 2: Find Numbers that Satisfy the Conditions

We need to find two numbers that multiply to 9 and add to -6. The numbers -3 and -3 satisfy these conditions because \((-3) imes (-3) = 9\) and \((-3) + (-3) = -6\).

Step 3: Write the Factored Form

Since both numbers are -3, we can express the trinomial as a square of a binomial. Thus, the factored form of \(x^2 - 6x + 9\) is \((x - 3)(x - 3)\) or \((x - 3)^2\).

Step 4: Verify the Factorization

Expand \((x - 3)^2\) to ensure it equals the original expression. When you expand \((x - 3)(x - 3)\), you get \(x^2 - 3x - 3x + 9\), which simplifies to \(x^2 - 6x + 9\). This confirms the factorization is correct.

Key Concepts

Quadratic Trinomials

Quadratic trinomials are a special type of polynomial that consists of three terms. These expressions can be written in the standard form of \(ax^2 + bx + c\), where the highest exponent is 2. This is why they're called "quadratic," which relates to the term "square." In our example, the quadratic trinomial is \(x^2 - 6x + 9\). Here, the coefficients are: \(a = 1\) for the squared term, \(b = -6\) for the linear term, and \(c = 9\) for the constant. Recognizing this form is essential, as it sets the stage for finding two numbers that work in tandem to both add up to \(b\) and multiply to \(c\).
Understanding how quadratic trinomials function will aid in factorizing and solving them or manipulating their algebraic properties efficiently. Quadratics appear frequently in various areas of mathematics and are foundational for studies in algebra, calculus, and beyond. Identifying these trinomials sets you up to apply factorization techniques, a backbone of polynomial algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation signs. They form the building blocks of mathematics. In this context, they allow us to represent real-world scenarios and mathematical relations abstractly yet precisely. The expression \(x^2 - 6x + 9\) is an algebraic expression, constructed using a variable \(x\) and constants (-6 and 9).
The challenge with algebraic expressions lies in their simplification and manipulation. This simplification process includes operations such as addition, multiplication, and, critically, factorization. For quadratics, these operations aim to reduce the expression to its simplest form—often a multiplication of binomials—as seen with the factorization of our quadratic trinomial into \((x - 3)^2\).

• This simplification makes solving equations easier.
• It reveals insights about relationships between the numbers involved.
• It often shows more clearly the roots of the equations or solutions of the problem at hand.

Algebraic expressions' flexibility and wide-ranging utility make them invaluable tools in mathematics.

Polynomial Factorization

Polynomial factorization is a valuable technique in algebra for breaking down complex expressions into simpler, more manageable pieces. This process is akin to solving a puzzle, finding components that work together to represent the original form. For the given quadratic trinomial \(x^2 - 6x + 9\), factorization simplifies the expression to \((x - 3)^2\). Understanding factorization is crucial, as it allows us to:

• Simplify expressions for easier computation or comparison.
• Identify the roots of polynomials efficiently, by setting each factor to zero.
• Make further mathematical operations, like integration or differentiation, more straightforward.

The factorization involves finding pairs of numbers that multiply to the constant term \(c\) and add to the linear coefficient \(b\). In this example, both requirements are satisfied by the number -3, leading us to recognize the expression as a perfect square trinomial. Factorization often simplifies solving equations, providing insight into the behavior of polynomial functions and their relations.