Problem 55 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(60 t^{4}+230 t^{3}-40 t^{2}\)

Step-by-Step Solution

Answer

The expression factors to \(10t^2(6t - 1)(t + 4)\).

Step 1: Identify the Greatest Common Factor (GCF)

Look at each term of the expression \(60t^4 + 230t^3 - 40t^2\). Determine if there is a common factor in all the terms. The GCF of the coefficients \(60, 230,\text{ and } 40\) is 10, and the smallest power of \(t\) in all terms is \(t^2\). Thus, the GCF of the entire expression is \(10t^2\).

Step 2: Factor Out the GCF

Divide each term of the expression by the GCF \(10t^2\) and factor it out from the expression: \[60t^4 + 230t^3 - 40t^2 = 10t^2(6t^2 + 23t - 4).\]

Step 3: Factor the Quadratic Expression

Now focus on factoring the quadratic \(6t^2 + 23t - 4\). Use the technique of splitting the middle term or the factorization method. We look for two numbers that multiply to \(6 \times (-4) = -24\) and add to \(23\). These numbers are \(24\) and \(-1\).

Step 4: Split the Middle Term and Factor by Grouping

Rewrite the quadratic \(6t^2 + 23t - 4\) by splitting the middle term: \[6t^2 + 24t - t - 4.\] Now group the terms: \[(6t^2 + 24t) + (-t - 4).\] Factor each group: \[6t(t + 4) - 1(t + 4).\] Factor out the common binomial factor: \[(6t - 1)(t + 4).\]

Step 5: Combine the Results

The complete factorization of the original expression is the product of the GCF and the factorized quadratic. Thus, \(60t^4 + 230t^3 - 40t^2 = 10t^2(6t - 1)(t + 4).\)

Key Concepts

Greatest Common Factor

Factoring begins with finding the greatest common factor (GCF) of the terms in a polynomial expression. The GCF is the largest factor that divides each term of the polynomial without leaving a remainder.

To identify the GCF in a polynomial like \(60t^4 + 230t^3 - 40t^2\), follow these steps:

• First, examine the coefficients. For the numbers \(60\), \(230\), and \(40\), the GCF is found by identifying the largest number that evenly divides all three, which is \(10\).
• Next, look at the variable parts. In this expression, the smallest power of \(t\) common to all terms is \(t^2\).

Thus, the GCF of the entire expression is \(10t^2\). Recognizing and factoring out the GCF simplifies the given expression and is a critical first step in the factorization process.

Quadratic Expressions

A quadratic expression is a polynomial of degree 2, often in the form \(ax^2 + bx + c\). These expressions can usually be factored into two binomial expressions.

In the context of \(6t^2 + 23t - 4\), we aim to break this down further. Factoring quadratics involves rewriting it in a form where two binomials are multiplied to give the original expression. This is like reverse foil in algebra, seeking binomials whose product matches the quadratic.

The challenge is in finding pairs of numbers that both add to the middle coefficient and multiply to the product of the quadratic coefficient and the constant term. In our example, the numbers are \(24\) and \(-1\). Correctly identifying these numbers is essential for moving to the next factorization stage.

Middle Term Splitting

Middle term splitting is a technique used to factor quadratics when the usual methods are not straightforward. It involves expressing the middle term as a sum of two terms whose coefficients multiply to give the product of the coefficient of the quadratic term and the constant term.

Let's see how it applies to \(6t^2 + 23t - 4\):

• Identify two numbers that multiply to \(-24\) (the product of \(6\) and \(-4\)) and sum to \(23\) (the middle term's coefficient).
  These numbers are \(24\) and \(-1\).
• Rewrite the middle term, \(23t\), as \(24t - 1t\).

This transforms the quadratic to \(6t^2 + 24t - t - 4\). By grouping terms \((6t^2 + 24t) + (-t - 4)\) and factoring separately, further solutions can be reached until the expression is fully factored. Understanding this technique allows students to break down complex quadratic expressions into simpler components.