Problem 38 - College Algebra Essentials

Question

In Exercises \(35-46,\) determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}-14 x $$

Step-by-Step Solution

Answer

Constant that should be appended to the binomial is \(49\). The trinomial is \(x^2 - 14x + 49\) and its factorized form is \((x - 7)^2\).

Step 1: Identify missing term

The provided binomial is \(x^2 - 14x\). The term \(b^2\) is not present. The term \(2ab\) in a perfect square trinomial has the value \(14x\). Therefore, we have that \(2*x*b = 14x\). Solving for \(b\) we acquire \(b = 7\). Therefore, the missing term \(b^2\) which is \(7^2 = 49\). Thus, the constant that should be appended to the binomial in order to form a perfect square trinomial is 49.

Step 2: Write the trinomial

On adding the found constant to the binomial, we get the trinomial which is \(x^2 - 14x + 49\). This is our perfect square trinomial.

Step 3: Factor the trinomial

Factorizing a perfect square trinomial involves finding values of \(a\) and \(b\) that satisfies the equation. We already identified that \(a = x\) and \(b = 7\). We can thus write \(x^2 - 14x + 49\) as \(x - 7)^2\).

Key Concepts

Understanding Binomials

A binomial is a type of polynomial with exactly two terms. In simple terms, it's a mathematical expression that consists of two components linked by either a plus or a minus sign. For example, in the expression \(x^2 - 14x\), there are two terms: \(x^2\) and \(-14x\). Binomials are the building blocks for more complex polynomials, such as trinomials. The terms can be numbers or variables or a combination of both.

• The number of terms: always two.
• Structure: typically \(ax^n\) and \(bx^m\).
• Can be combined to form trinomials and higher polynomials.

This understanding helps us transform and manipulate expressions, especially when preparing to create perfect square trinomials.

Exploring Trinomials

A trinomial is an extension of a binomial and comprises three terms. For example, a trinomial like \(x^2 - 14x + 49\) is formed by adding a new term to a binomial. The structure of a trinomial is generally viewed as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.In the context of perfect square trinomials, every trinomial can be rewritten as the square of a binomial. When transformed into the form \((x \, \pm \, k)^2\), it reveals important insights into its roots and factors.

• Three terms: typically arranged as a quadratic equation.
• Can be factored into a binomial expression.
• Useful for solving quadratic equations and understanding complex polynomial behavior.

The Art of Factoring

Factoring is a mathematical process used to break down expressions into products of simpler expressions. For example, the trinomial \(x^2 - 14x + 49\) can be factored into \((x - 7)^2\).The goal of factoring is often to simplify expressions or solve equations. It's a fundamental skill in algebra and helps in understanding the structure and roots of polynomials. Here are some key points about factoring:

• Breaks down polynomials into simpler, manageable parts.
• Can reveal solutions to equations by isolating roots.
• Often used to find greatest common factors or roots of quadratic equations.

By factoring expressions, we can solve for unknowns and understand deeper mathematical relationships.

The Role of the Constant Term

In a trinomial, the constant term is the term that doesn't contain any variables, represented by \(c\) in the general form \(ax^2 + bx + c\). For instance, in the trinomial \(x^2 - 14x + 49\), the number 49 is the constant term.Adding an appropriate constant term to a binomial can transform it into a perfect square trinomial. The constant term influences both the position of the parabola graph and the roots of the equation.

• Determines the vertical position of the graph.
• Affects the factors and solutions of quadratic equations.
• Important when completing the square to form a perfect square trinomial.

Understanding the role of the constant term aids in creating balance and precision within polynomial equations.