Problem 9 - Algebra Form and Function

Question

Give the constant term, \(a_{0}\). $$ (3 t+1)(2 t-1) $$

Step-by-Step Solution

Answer

Answer: -1

Step 1: Identify the expressions to be multiplied

We are given the two expressions \((3t + 1)\) and \((2t - 1)\) to multiply.

Step 2: Perform the multiplication using the distributive property (FOIL)

Multiply the two binomial expressions using the FOIL method (First, Outer, Inner, and Last terms): 1. Multiply the First terms: \((3t)(2t)\). 2. Multiply the Outer terms: \((3t)(-1)\). 3. Multiply the Inner terms: \((1)(2t)\). 4. Multiply the Last terms: \((1)(-1)\).

Step 3: Combine the results of the multiplication

Add the results from Step 2: 1. \((3t)(2t) = 6t^2\). 2. \((3t)(-1) = -3t\). 3. \((1)(2t) = 2t\). 4. \((1)(-1) = -1\). So the combined expression is \(6t^2 - 3t + 2t - 1\).

Step 4: Simplify the expression

Combine like terms: \(6t^2 - 3t + 2t - 1 = 6t^2 - t - 1\).

Step 5: Identify the constant term \(a_{0}\)

In the simplified expression \(6t^2 - t - 1\), the constant term is \(-1\). So, \(a_{0} = -1\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply an expression by distributing a factor across terms within parentheses. This property is a cornerstone in dealing with algebraic expressions and plays a vital role when solving equations.
When you apply the distributive property, you're essentially ensuring that each term inside the parentheses is multiplied by the factor outside. In mathematical terms, for expressions like \((a + b)(c + d)\), you distribute such that every term in the first parenthesis is multiplied by every term in the second parenthesis:

• Multiply \( a \times c \)
• Multiply \( a \times d \)
• Multiply \( b \times c \)
• Multiply \( b \times d \)

This approach is systematic and ensures that all combinations are accounted for, making the problem-solving process organized and reliable. It leads us directly into the FOIL method for binomials, providing a structured way of handling these calculations.

FOIL Method

The FOIL method is a handy mnemonic device that specifically applies the distributive property to binomials. FOIL stands for First, Outer, Inner, and Last, referring to the sequence in which you multiply terms when dealing with expressions like \((3t + 1)(2t - 1)\).
Here's how it works:

• First: Multiply the first terms in each binomial, i.e., \((3t)(2t) = 6t^2\).
• Outer: Multiply the outer terms, i.e., \((3t)(-1) = -3t\).
• Inner: Multiply the inner terms, i.e., \((1)(2t) = 2t\).
• Last: Multiply the last terms, i.e., \((1)(-1) = -1\).

The result of applying FOIL to our binomials gives us each piece of the product expression: \[6t^2 - 3t + 2t - 1\].
It's important to remember that while FOIL is specific to the multiplication of two binomials, its core principle of using the distributive property can be applied to multiplying any polynomials.

Simplifying Expressions

Once we've multiplied the terms using the distributive property or FOIL method, it's crucial to simplify the resulting expression.
Simplifying means combining like terms to present the expression in its simplest form.
In our expression \(6t^2 - 3t + 2t - 1\), we see that \(-3t\) and \(+2t\) are like terms. This means they are similar in terms of variables and exponents and can be combined:

• Combine \(-3t + 2t\) to get \(-t\)

This simplification process reduces the expression to \(6t^2 - t - 1\).
Finally, by simplifying, we easily identify the constant term \(a_0\), which is the term without any variable. In this case, it is \(-1\), the fixed number in the expression. Understanding how to simplify expressions efficiently is essential for solving algebraic problems correctly and making complex calculations more manageable.