Problem 10 - Elementary and Intermediate Algebra

Question

CONCEPTS Use multiplication to determine whether \((3 t-1)(5 t-6)\) is the correct factorization of \(15 t^{2}-19 t+6\)

Step-by-Step Solution

Answer

The factorization is incorrect; the correct middle term should yield \(-19t\) when expanded.

Step 1: Expand the Expression

To check the factorization, start by expanding the expression \((3t-1)(5t-6)\) using the distributive property (also known as FOIL for binomials). The expression expands to: \\( (3t * 5t) + (3t * -6) + (-1 * 5t) + (-1 * -6) \). \Simplify each term to get the expression\(15t^2 - 18t - 5t + 6\).

Step 2: Combine Like Terms

Combine the like terms in the expanded expression found in step 1. The terms to combine are \(-18t\) and \(-5t\): \ \(-18t - 5t = -23t\). \Thus, the expanded expression becomes \(15t^2 - 23t + 6\).

Step 3: Compare with the Original Polynomial

Compare the expanded expression from Step 2 \(15t^2 - 23t + 6\) with the original polynomial \(15t^2 - 19t + 6\). \Notice that the middle term \(-23t\) does not match the middle term \(-19t\) of the original polynomial.

Key Concepts

Distributive Property

The Distributive Property is a foundational concept in algebra that makes it possible to simplify expressions and solve equations. This property states that multiplying a sum by a number gives the same result as multiplying each addend separately and then adding the products. For instance, in more formal terms, this can be described as:

• \[ a(b + c) = ab + ac \]

In the context of the polynomial factorization problem with \((3t-1)(5t-6)\), the Distributive Property helps us understand how to expand these binomials. Here’s how it works:

• First, \[ 3t \times 5t = 15t^2 \]
• Second, \[ 3t \times -6 = -18t \]
• Next, \[ -1 \times 5t = -5t \]
• Finally, \[ -1 \times -6 = 6 \]

These calculations demonstrate how the Distributive Property expands each part of the binomial, leading to a new array of terms, which then requires combining any like terms.

FOIL Method

The FOIL Method is a technique specifically used to multiply two binomials. FOIL stands for First, Outer, Inner, and Last, which indicates the order in which you multiply the terms in the binomials.
To apply the FOIL Method to \((3t-1)(5t-6)\), follow these steps:

• First: \[ 3t \times 5t = 15t^2 \]
• Outer: \[ 3t \times -6 = -18t \]
• Inner: \[ -1 \times 5t = -5t \]
• Last: \[ -1 \times -6 = 6 \]

Each step corresponds to multiplying a specific pair of terms from the two binomials. After using FOIL, you get the result:\[ 15t^2 - 18t - 5t + 6 \].
Using FOIL simplifies the process of seeing all products clearly, as it lays out each multiplication step.

Combining Like Terms

Once the expression is expanded using either the Distributive Property or FOIL Method, the next step is to simplify it by combining like terms. This means finding terms that have the same variable component.
For the polynomial \(15t^2 - 18t - 5t + 6\), like terms are crucial. The terms \(-18t\) and \(-5t\) both have the same variable \(t\), which makes them like terms.
By combining these:

• \[ -18t - 5t = -23t \]

This operation reduces the expanded polynomial to \(15t^2 - 23t + 6\). Recognizing and combining like terms are essential skills in polynomial factorization, simplifying expressions, and verifying factorization results.