Problem 23 - Beginning and Intermediate Algebra

Question

Factor completely. $$36 t^{2}-60 t u+25 u^{2}$$

Step-by-Step Solution

Answer

The factored form of the given polynomial is \((6t - 5u)^2\).

Step 1: 1. Finding the Greatest Common Factor (GCF)

The first step is to identify the greatest common factor (GCF) that can be factored out from all of the coefficients. In this case, the GCF of 36, -60, and 25 is 1. Since 1 is the GCF, we cannot factor out any common factor from the expression.

Step 2: 2. Check if it is any special type of quadratic trinomial

We can check if the given polynomial is a perfect square trinomial or a difference of two squares. The general form of a perfect square trinomial is: \(a^2 + 2ab + b^2\) or \(a^2−2ab+b^2\). Let's see if the given polynomial matches this pattern. In this case, \(a^2 = 36t^2\) \(b^2 = 25u^2\) \(2ab = 2(6t)(5u) = 60tu\) The given expression is: \(36t^2−60tu+25u^2\) Now, we can see that it fits the pattern of a perfect square trinomial in the form of \(a^2−2ab+b^2\).

Step 3: 3. Factoring the expression

Since the given polynomial fits the pattern of a perfect square trinomial, we can rewrite the polynomial as a binomial squared. In this case, the binomial will be: \((a - b)^2\) We have: \(a^2 = 36t^2\) which implies \(a = 6t\) \(b^2 = 25u^2\) which implies \(b = 5u\) So, substituting the values of a and b in the expression \((a - b)^2\), we get: \((6t - 5u)^2\) Hence, the factored form of the given polynomial is \((6t - 5u)^2\).

Key Concepts

Understanding Quadratic Trinomials

Quadratic trinomials are algebraic expressions consisting of three terms, where the highest degree of the variable is two. These expressions usually take the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients.
Recognizing a quadratic trinomial is crucial in algebra, as it helps in executing various factoring techniques to simplify expressions or solve equations.
In our example, the expression is \(36t^2 - 60tu + 25u^2\), where each component aligns with the typical structure of a quadratic trinomial:

• \(a = 36t^2\) is the quadratic term.
• \(b = -60tu\) is the linear term.
• \(c = 25u^2\) is the constant or simplified as "a term with no t-variable."

Recognizing these terms assists in identifying the correct factoring method, setting the foundation for simplifying complex algebraic expressions.

Exploring Factoring Techniques

Different factoring techniques are used depending on the form and characteristics of the algebraic expression. One common method is factoring by grouping or looking for patterns such as perfect square trinomials. A perfect square trinomial has the form:\((a^2 + 2ab + b^2)\) or \((a^2 - 2ab + b^2)\).
In our example, \(36t^2 - 60tu + 25u^2\) fits the form of a perfect square trinomial \((a^2 - 2ab + b^2)\).
Here's how you identify the pattern and factor:

• Identify \(a^2\) and \(b^2\): This gives \(a = 6t\) and \(b = 5u\).
• Verify that the middle term equals \(2ab\): In this case, \(2(6t)(5u) = 60tu\).
• Match it to the form \((a - b)^2\) because of the subtraction in \(-60tu\).

Using these steps can help you factor the expression correctly, allowing you to rewrite \(36t^2 - 60tu + 25u^2\) as \((6t - 5u)^2\).
Factoring techniques simplify expressions and aid in solving equations.

Understanding Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators, such as addition or subtraction. They are the backbone of algebra and are used to formulate and solve mathematical problems.
In our exercise, the expression is \(36t^2 - 60tu + 25u^2\). This expression includes:

• Variables: \(t\) and \(u\)
• Coefficients: \(36\), \(-60\), and \(25\)
• Operators: subtraction in between the terms

Different expressions can represent relationships or functions, and understanding them is crucial for solving equations efficiently.
Grasping how to manipulate these expressions, such as by factoring them, is key, especially in turning complex problems into simpler forms.
It's also important to note the role of each component: variables represent unknowns, coefficients provide the multiplicative factor of these variables, and operators define the mathematical operation performed between the terms.
Thus, mastering algebraic expressions enables mathematicians and students alike to tackle various mathematical challenges effectively.