Problem 52 - Algebra 1: Concepts and Skills

Question

Factor the expression completely. $$ 2 t^{3}-98 t $$

Step-by-Step Solution

Answer

The complete factorization of the given expression is \(2t (t - 7)(t + 7)\).

Step 1: Identifying The Common Factor

The first step in factorization is to identify the common factor for all the terms in the expression. Take a look at \(2 t^3 - 98 t\). The factors of \(2 t^3\) are 2, \(t\), \(t\), and \(t\). The factors of \(-98 t\) are 2, 7, 7, \(-1\), and \(t\). The common factor in these two expressions is \(2t\).

Step 2: Factor out The Common Factor

Factor out the common factor from the entire expression. You end up with: \(2t (t^2 - 49)\).

Step 3: Look for Further Factorization

Look at the expression \((t^2 - 49)\) and recognize it as a difference of two squares. A difference of squares can be factored into \(a^2 - b^2 = (a - b)(a + b)\), where \(a = t\) and \(b = 7\).

Step 4: Factorize the Difference of Squares

Apply the rule of difference of squares to get the complete factorization of the expression: \(2t (t - 7)(t + 7)\).

Key Concepts

Common Factor

Understanding the idea of a common factor is crucial to mastering algebra. When factoring polynomials, the common factor refers to a term that is shared by all parts of the algebraic expression. Think of it as the largest piece of the mathematical puzzle that fits into each part of the equation.

For instance, in the expression 2t^3 - 98t, both terms share a factor of 2t. This is what we extract first to simplify the polynomial. By factoring out this common element, we're essentially dividing each term of the polynomial by 2t, which simplifies the expression and lays the groundwork for further factorization that could reveal more intricate patterns hidden within the algebraic expression.

Difference of Squares

The concept of difference of squares is an elegant shortcut in polynomial factorization, applicable when a polynomial can be expressed as the difference between two perfect squares. In algebra, a perfect square is an expression that can be squared to produce the target value.

A classic example is a^2 - b^2, which factors into (a + b)(a - b). This identity saves time and simplifies expressions significantly. When applying this to our problem, we identify t^2 - 49 as such a case, where t^2 and 49 are both perfect squares. Recognizing this pattern allows us to factor it instantly into (t - 7)(t + 7), thus revealing the hidden structure within the polynomial.

Polynomial Factorization

At its core, polynomial factorization is the process of breaking down a complex algebraic expression into a product of simpler factors. It's like disassembling a machine to understand its components. This process is not just about simplification; it's about revealing the underlying properties and potential solutions to equations.

Our original problem involves taking the polynomial 2t^3 - 98t and breaking it down step-by-step. Initially, by identifying and factoring out a common factor, and subsequently by using special patterns such as the difference of squares. The goal is to rewrite the polynomial as a product of binomials or other polynomials that, when multiplied together, give you the original expression.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like t in our example), and operators (like + and -). Algebraic expressions are the sentences of algebra and convey quantities and relationships. When we perform operations like factorization on these expressions, we're not changing their value – we're merely rewriting them in a different form to reveal additional insights or to simplify calculations.

Understanding how to handle these expressions is essential when solving for unknowns, simplifying equations, and in higher math, interpreting graphs. By becoming proficient in spotting common factors and recognizing patterns like the difference of squares, you'll be well-equipped to tackle a wide range of algebraic challenges.