Problem 10 - Introductory Algebra for College Students

Question

Complete the square for binomial. Then factor the resulting perfect square trinomial. \(x^{2}+\frac{1}{3} x\)

Step-by-Step Solution

Answer

The binomial \(x^{2}+\frac{1}{3} x\) when completed the square becomes the perfect square trinomial \(x^{2}+\frac{1}{3} x + \frac{1}{36}\) which can be factored as \( (x + \frac{1}{6})^{2} \).

Step 1: Represent the Binomial in Standard Form

Let's look at the binomial expression \(x^{2}+\frac{1}{3} x\). The binomial is already arranged in standard form.

Step 2: Complete the Square

To complete the square, add \( (\frac{b}{2a})^{2} \) to the binomial expression. In this case, \(a = 1\) and \(b = \frac{1}{3}\). So the added constant will be \((\frac{\frac{1}{3}}{2 * 1})^{2}\), which equals \( \frac{1}{36}\).Therefore, the binomial expression becomes a trinomial by adding this constant, which now becomes \(x^{2}+\frac{1}{3} x + \frac{1}{36} \).

Step 3: Factor the Perfect Square Trinomial

The perfect square trinomial \(x^{2}+\frac{1}{3} x + \frac{1}{36}\) can now be factored. This trinomial can be written as \( (x + \frac{1}{6})^{2} \).

Key Concepts

Factoring Trinomials

Factoring trinomials is a vital skill in algebra. It involves rewriting a polynomial expression as a product of binomial expressions. You encounter trinomials in the form of \(ax^2 + bx + c\). To factor them, you look for two numbers that multiply to \(ac\) and add up to \(b\). This technique simplifies solving algebraic equations, making it much easier to identify the roots.

• First, you should identify the values of \(a\), \(b\), and \(c\).
• Then, find two numbers that multiply to \(ac\) and sum to \(b\).
• Split the middle term using these numbers, if necessary, and factor by grouping.

In some cases, the trinomial might already be a perfect square, which further simplifies the process. Knowing how to factor trinomials is also essential when completing the square or solving quadratic equations.

Perfect Square Trinomial

A perfect square trinomial is a special type of trinomial formed by squaring a binomial.Imagine you have \((x + 1)^2\) expanded to form a trinomial: \(x^2 + 2x + 1\). This can be represented as a perfect square trinomial because it can be factored back into \((x + 1)^2\).

• To recognize a perfect square trinomial, check if the first and last terms are perfect squares.
• Verify if the middle term is twice the product of the roots of the first and last terms.

These are not ordinary trinomials because they fit a particular pattern, \(a^2 + 2ab + b^2\), which factors into \((a + b)^2\). When completing the square, the goal is often to create a perfect square trinomial, making it simpler to factor and solve.

Binomial Expression

A binomial expression is a polynomial with exactly two terms. An example is \(x + y\). Binomials are fundamental building blocks in algebra. They play a critical role in forming higher-degree polynomials and discovering relationships between variables.

• Identify binomials by finding expressions with precisely two distinct terms.
• Binomials can be added, subtracted, multiplied, and divided, just like any polynomial expressions.

They can often be manipulated into special forms such as perfect squares or difference of squares. Completing the square, for instance, starts with a binomial and adds another term to make a trinomial.