Problem 11 - Introductory Algebra for College Students

Question

Complete the square for binomial. Then factor the resulting perfect square trinomial. \(x^{2}-\frac{4}{3} x\)

Step-by-Step Solution

Answer

The factored form of perfect square trinomial is \((x-\frac{2}{3})^{2}\).

Step 1: Adding the squared half of the coefficient of \(x\)

The coefficient of \(x\) in the binomial is \(-\frac{4}{3}\). Its half is \(-\frac{4}{3} \div 2 = -\frac{2}{3}\). The square of \(-\frac{2}{3}\) is \((-(-\frac{2}{3}))^{2} = \frac{4}{9}\). So, the binomial becomes \(x^2 -\frac{4}{3}x + \frac{4}{9}\).

Step 2: Factoring the perfect square trinomial

The trinomial obtained in the previous step, \(x^2 -\frac{4}{3}x + \frac{4}{9}\), is now a perfect square trinomial. As it's a perfect square, it can be factored to \((x-\frac{2}{3})^{2}\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that can be expressed as the square of a binomial. This means it takes the form

• \( (ax + b)^2 = a^2x^2 + 2abx + b^2 \),

where each part of the trinomial originates from squaring the binomial. In the context of the exercise, we started with the binomial \(x^2 - \frac{4}{3} x \). To transform it into a perfect square trinomial, we added \(\frac{4}{9}\)—the square of half the `x` coefficient (-\(\frac{2}{3}\)). Without this step, the expression wouldn't be a perfect square trinomial.Creating a perfect square trinomial allows us to factorize the quadratic expression easily, which is useful for solving equations and understanding the expression's properties.

Factoring Trinomials

Factoring trinomials involves expressing them as the product of two binomials. This process is crucial for simplifying expressions and solving quadratic equations. Generally, for a trinomial given as \(ax^2 + bx + c\), the goal is to write it in the form

• \((px + q)(rx + s)\),

where `p`, `q`, `r`, and `s` are numbers that when multiplied and combined reproduce the original trinomial.In the exercise, after completing the square, we had the trinomial \(x^2 - \frac{4}{3}x + \frac{4}{9}\). Known as a perfect square trinomial, it directly factors into \((x - \frac{2}{3})^2\). This demonstrates the simplicity of factoring perfect square trinomials—they always break down into identical binomials squared. This is a powerful algebraic tool for calculations and simplifications.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operations. They form the basis for more complex mathematical concepts and algebraic manipulations. They can be as simple as a single number or variable, or more complex, like the polynomial used in this exercise.Understanding algebraic expressions includes knowing how to handle each component, like coefficients (numbers multiplying the variables), variables themselves, and constant terms. Recognizing patterns such as like terms and the principles behind operations such as factoring is vital.In our example, the expression \(x^2 - \frac{4}{3}x\) is an algebraic expression with terms involving both variables and coefficients. Completing the square is an exercise in manipulating this expression into a more useful form. Such transformations are important for solving equations and making sense of complex mathematical problems.