Problem 13 - Algebra A Combined Function

Question

Factor each trinomial completely. $$ x^{2}-16 x+64 $$

Step-by-Step Solution

Answer

\((x-8)^2\)

Step 1: Identify the Elements of the Trinomial

The given trinomial is in the form of a quadratic expression, \( ax^2 + bx + c \), where \( a = 1 \), \( b = -16 \), and \( c = 64 \).

Step 2: Determine the Factors of 'c'

We need to find two numbers that multiply to \( c = 64 \) and add to \( b = -16 \). Observe that 8 and 8 satisfy both conditions because \( 8 \times 8 = 64 \) and adding them gives \( 8 + 8 = 16 \). Since \( b \) is negative, we use \(-8\) and \(-8\): \((-8) + (-8) = -16\).

Step 3: Write the Factored Form

Since \( -8 \) is repeated, the trinomial \( x^2 - 16x + 64 \) factors to \((x - 8)(x - 8)\), or \((x - 8)^2\) since both binomials are the same.

Key Concepts

Quadratic Expressions

Quadratic expressions are a type of polynomial with the highest degree of 2. They are typically in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. The "quadratic" term originates from the Latin word "quadratus," referring to the square. What distinguishes quadratic expressions is the presence of the squared variable, making it foundational in algebra.
Understanding how to manipulate these expressions is essential for solving various mathematical problems, particularly those involving motion and area calculations.
A key goal in working with quadratics is often to rewrite them in factored form, which simplifies solving equations.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into simpler polynomials (called "factors") that, when multiplied together, give the original polynomial. This process is like reverse expanding in multiplication.
Factoring polynomials is a fundamental skill because it helps in solving polynomial equations by setting each factor equal to zero and solving for the variable. By reducing a complex expression into simpler parts, factoring highlights solutions and interesting properties of the polynomial.
For instance, in factoring, you can identify roots or solutions to equations, which is crucial in fields like engineering and physics.

Trinomial Factoring

Trinomial factoring is a subset of polynomial factoring focused on expressions of the form \( ax^2 + bx + c \). The goal is to rewrite the trinomial as a product of two binomials.
Given the exercise \( x^2 - 16x + 64 \), we first identify the coefficients \( a = 1 \), \( b = -16 \), and \( c = 64 \). The challenge is to find two numbers that both multiply to \( c \) and add to \( b \).
In this case, both numbers are \(-8\), since \((-8) \times (-8) = 64\) and \((-8) + (-8) = -16\). This process results in the binomial expression \((x - 8)^2\), showcasing how trinomial factoring reveals the expression's structure and solutions.

Factorization Techniques

Factorization techniques are strategies used to simplify polynomial expressions into their factors. These techniques vary depending on the expression type, but common methods include:

• Identifying common factors: Look for terms that share a common factor and factor them out.
• Using special formulas: Such as the difference of squares \((a^2 - b^2 = (a - b)(a + b))\).
• Splitting the middle term: Break the middle term into two terms to facilitate factoring by grouping.

In the exercise provided, we used the splitting middle term technique to factor \( x^2 - 16x + 64 \). The number pair \(-8\) and \(-8\) made it straightforward since both conditions of multiplying to 64 and adding to \(-16\) were met. This technique makes the trinomial factoring process more manageable, especially for beginners learning the intricacies of polynomial expressions.