Problem 72 - Beginning and Intermediate Algebra

Question

Factor completely. $$14 s^{2}(t-13)^{5}+69 s(t-13)^{5}+27(t-13)^{5}$$

Step-by-Step Solution

Answer

The completely factored expression is \((t-13)^{5}(14s^2 + 69s + 27)\).

Step 1: Identify the common factor

In this case, all three terms have the common factor \((t-13)^{5}\). So, we can factor this out of the entire expression.

Step 2: Factor out the common factor

Using the distributive property, we factor out \((t-13)^{5}\), as follows: $$14 s^{2}(t-13)^{5}+69 s(t-13)^{5}+27(t-13)^{5} \Rightarrow$$ $$(t-13)^{5}(14s^2 + 69s + 27)$$ Now that we have factored out the common factor, the expression has been simplified. But the remaining quadratic expression may be factorable as well.

Step 3: Analyze the quadratic

The quadratic expression remaining after factoring the \((t - 13)^{5}\) is: $$(14s^2 + 69s + 27)$$ Since all the coefficients are integers, we can check if there are factors of 27 that add up to 69 when multiplied by 14.

Step 4: Attempt to factor the quadratic

After analyzing the factors, unfortunately, there is no combination of factors of 27 that will result in the middle term being 69 when multiplied by 14. Therefore, this quadratic is not factorable. So, the final completely factored expression is:

Step 5: Write the final factored expression

$$(t-13)^{5}(14s^2 + 69s + 27)$$

Key Concepts

Understanding Algebra and Its Importance

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols to solve problems. It teaches you how to handle equations, variables, and algebraic expressions. Algebra is essential because it forms the basis for many other areas of math, science, and engineering.

• Algebraic expressions are combinations of numbers, variables, and arithmetic operations.
• Algebra helps you translate real-world situations into mathematical models.

Through learning algebra, you gain valuable problem-solving skills. It enables you to approach complex problems systematically and find solutions efficiently.

Exploring Quadratic Expressions

Quadratic expressions are polynomials that include a term with a variable raised to the second power, typically written in the form of \(ax^2 + bx + c\). These expressions are crucial in algebra because they arise in various real-world applications, such as calculating areas, projectile motion, and optimization problems.

• The general form of a quadratic expression is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
• The quadratic term \(ax^2\) makes these expressions "quadratic," as "quad" refers to a square - the power of two.

Quadratic expressions can often be factored into simpler expressions, making it easier to solve equations or analyze functions. However, not all quadratics are easily factorable, as seen in this exercise.

The Distributive Property in Action

The distributive property is a key algebraic tool that allows you to multiply a single term by each term within a parenthesis. It's written as \(a(b + c) = ab + ac\). This property is particularly useful when factoring expressions. It helps break down expressions into simpler forms, making calculations more manageable.

• The distributive property allows you to "distribute" multiplication over addition or subtraction.
• Using this property, you can extract a common factor from an expression, thus simplifying it.

In the original exercise, the distributive property was applied to factor out the common term \((t-13)^5\) from each part of the expression.

Identifying and Using a Common Factor

A common factor is a term that is shared by two or more terms in an expression. Discovering a common factor is a crucial step in simplifying expressions, especially when working with polynomials. Factoring by identifying the common factor involves:

• Recognizing the common term or expression in multiple parts of an equation.
• Extracting this factor to simplify the expression.

In our exercise, \((t-13)^5\) was the common factor in all terms. By factoring it out, the expression was greatly simplified, leaving a quadratic expression behind. This approach is often the first step in the factoring process and is essential for reducing complexity and making mathematical work more streamlined and efficient.