Problem 53 - Elementary and Intermediate Algebra

Question

Factor out the GCF. $$ t^{4}+t^{3}+2 t^{2} $$

Step-by-Step Solution

Answer

The factored form is \(t^2(t^2 + t + 2)\).

Step 1: Identify the GCF

To factor out the greatest common factor (GCF) from the polynomial, begin by identifying the smallest power of any common variable among the terms. In the expression \(t^4 + t^3 + 2t^2\), each term has the variable \(t\). The smallest power of \(t\) present in all terms is \(t^2\). Hence, the GCF of the polynomial is \(t^2\).

Step 2: Factor out the GCF

Once we have identified the GCF as \(t^2\), we factor \(t^2\) out of the polynomial. This means dividing each term by \(t^2\): \[t^4 + t^3 + 2t^2 = t^2(t^2) + t^2(t) + t^2(2)\].

Step 3: Write the factored expression

Now, we write the original polynomial as a product of the GCF and the simplified polynomial inside the parentheses. The expression simplifies to: \[t^2(t^2 + t + 2)\]. Thus, the factored form of the polynomial is \(t^2(t^2 + t + 2)\).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, often abbreviated as GCF, is an essential concept in algebra, especially when working with polynomial expressions. It refers to the largest factor that divides two or more numbers or terms in an expression. In simple terms, it's the biggest number or term that can fit into every term of a polynomial without leaving a remainder. To find the GCF of polynomial terms:

• Look for common factors in terms of numbers and variables.
• Identify any shared variable parts and pick the smallest exponent among them.

Let's consider the polynomial expression given: \[t^4 + t^3 + 2t^2\]. Each term has the variable \(t\), and the smallest exponent of \(t\) across all terms is 2 (i.e., \(t^2\)). Hence, \(t^2\) is the GCF. Finding and factoring out the GCF is an essential skill, as it simplifies complex expressions and is the first step in factoring more complicated polynomials.

Polynomial Expressions

Polynomial expressions are mathematical expressions consisting of variables and coefficients. They are constructed using operations of addition, subtraction, multiplication, and non-negative integer exponents on variables. For example, the expression \(t^4 + t^3 + 2t^2\) is a polynomial.Key aspects of polynomial expressions include:

• Terms: Each part of the expression separated by a plus or minus sign (e.g., \(t^4\), \(t^3\), and \(2t^2\)).
• Degree: The highest power of the variable in the polynomial (e.g., 4 in the expression \(t^4\)).
• Coefficients: Numbers multiplying the variables (e.g., 1 in \(t^4\), 1 in \(t^3\), and 2 in \(2t^2\)).

Understanding what makes up a polynomial helps in identifying factors and distinguishing between different types of polynomial expressions. Factoring polynomials often starts with identifying the structure and components of the expression.

Algebraic Expressions

Algebraic expressions are a broader category encompassing polynomials and other forms of expressions consisting of variables and constants. Unlike polynomials, algebraic expressions can include fractions, square roots, and other types of numbers. In algebra, simplifying expressions is a core activity. Here are steps to work with algebraic expressions:

• Identify like terms: Terms with the same variable part (e.g., like terms in \(t^4\) and \(t^3\) both have 't').
• Simplify terms: Combine like terms or factor common terms out, as we did with \(t^4 + t^3 + 2t^2\), factoring out \(t^2\).
• Perform operations carefully: Respect the order of operations (i.e., parentheses, exponents, multiplication and division, addition and subtraction - PEMDAS).

By mastering algebraic expressions, students not only simplify complex problems but also strengthen their ability to analyze and solve equations systematically.