Problem 36 - 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. $$ x^{2}+16 x $$

Step-by-Step Solution

Answer

The constant that should be added is 64, making the perfect square trinomial \(x^{2} + 16x + 64\), and the factored form is \((x+8)^{2}\).

Step 1: Find the constant to complete the trinomial

To find the constant term that completes the trinomial, the coefficient of the 'x' term, in this case 16, needs to be divided by 2 and then squared. Therefore, \( (16/2)^{2} = 8^{2} = 64 \).

Step 2: Write the perfect square trinomial

By adding the constant found in Step 1 to the given binomial, a perfect square trinomial can be written. Therefore, our trinomial is \( x^{2} + 16x + 64 \).

Step 3: Factoring the trinomial

Finally, factor the resulting perfect square trinomial. Using the formula \((a^{2} + 2ab + b^{2} = (a + b)^{2})\), for our trinomial where a is 'x' and b is '8', the factored form of the trinomial is \((x+8)^{2}\).

Key Concepts

Factoring

Factoring is a fundamental concept in algebra that involves breaking down more complex expressions into simpler components. When we factor expressions, we aim to express the original equation as a product of its factors.

In the context of our problem, factoring is applied to a perfect square trinomial. A trinomial is a polynomial with three terms. When it comes to a perfect square trinomial, there is a specific pattern that allows us to factor it efficiently. This pattern is:

• If you have a trinomial of the form \[a^2 + 2ab + b^2\], it can be factored as \[(a+b)^2\].

The exercise provided a binomial, \[x^2 + 16x\], to which we added a specific constant to complete the square and transform it into a perfect square trinomial. Once the trinomial was formed, it was straightforward to factor it into \[(x+8)^2\].

Factoring using patterns like this makes solving polynomial equations much easier, simplifying not just the analysis but also the calculations required in subsequent steps.

Quadratic Expression

A quadratic expression is an algebraic expression that involves a squared term as the highest degree. For instance, the expression \[x^2 + 16x\] is a quadratic expression. It comprises three parts:

• The quadratic term, \(x^2\).
• The linear term, \(16x\).
• And a constant that completes the expression into a perfect square trinomial.

In this exercise, we aimed to turn the expression into a perfect square trinomial by determining the appropriate constant to add. This is done by completing the square, which involves manipulating the quadratic expression to make it factorable into a single binomial squared.

Understanding quadratic expressions is crucial because they form the foundation for solving quadratic equations, which frequently appear in both academic and real-world scenarios.

Binomial

A binomial is a polynomial made up of exactly two terms. These terms are usually separated by an addition or subtraction sign. An example of a binomial is \[x^2 + 16x\] from the exercise. The challenge often involves converting or extending binomials into other forms to simplify and solve expressions or equations.

When dealing with binomials in algebra, especially within the context of creating a perfect square trinomial, the goal is to add a specific constant derived from the existing terms so that the new expression can be factored neatly. This involves calculating the correct term to add by following the square completion process: divide the coefficient of the middle term by two, square it, and add it to the binomial.

By understanding binomials, you set the stage for more complex algebraic manipulations, ultimately aiding in solving equations more efficiently.